







2color combinations 
20180105 

From Sabrina: How many 2color (pair) combinations can be made from 12 individual colors... is it (11x12) or (optional): (11x12)x2? Answered by Penny Nom. 





Retiring athletes numbers 
20171218 

From Robin: I am wondering if you can explain in layman terms how to figure out how many different ways the numbers 0099 can be arranged? The discussion that sparked this question revolved around the practice of retiring athletes numbers and how many actual number combinations there are. So if I were to use 00, 01, 02 03, 04, 05, 06, 07, 08, 09, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ....99. I say there are only 109 different numbers. I'm being told I'm wrong that there are way more than that. Please help. Thank you, Robin Answered by Penny Nom. 





10 digit permutations 
20171124 

From Ed: HI my question..
numbers1,2,3,4,5,6,7,8,9,0 how many combinations are there
example: 518, 269, 3470
example2:815,972, 3460
example3: 185,047,2369 Answered by Penny Nom. 





20 things taken 8 at a time 
20160530 

From Jessie: I need all of the 8 digit combinations using the numbers 120.
i.e 1 2 3 4 5 6 7 8, 2 3 4 5 6 7 8 9 like wise
no number should be repeated in one combination.
Thanks
Jessie Answered by Penny Nom. 





Arrangements of the letters STRANGE 
20150822 

From Ravi: In how many ways can the letters of the word STRANGE be arranged to that the vowels occupy only the old places. Answered by Penny Nom. 





4 digit strings using the digits 1, 2, 3 and 4 
20150614 

From Gary: What is the number of possible combinations using the following fact pattern?
A. Each combination must have 4 numbers.
B. The possible numbers to choose from are 1, 2, 3, 4
C. The numbers can be included in the combination more than once. For example, it can be 1111.
Is it possible to generate a schedule of the possibilities? Answered by Penny Nom. 





Four digit phone numbers 
20140225 

From Tom: Hi, I'm an online math teacher, working on lessons for my students.
This question is in their book and I can't do it...! Help!
It has to do with phone numbers, and in this problem,
we are only dealing with
the last 4 digits of the number.
It asks how many possible numbers we have
if at least one digit repeats in the last 4 digits of the number. Answered by Penny Nom. 





Some 3digit combinations 
20140223 

From ivana: How many possible 3digit combinations can be made of the numbers 123456?
None of the numbers can repeat in one combination. I used to know a formula but have been out of school for some time and cannot remember. Answered by Penny Nom. 





6 digit permutations from {149} 
20121227 

From ange: how many six digits numbers permutations can be formed with the set {149} if no digit is used twice in each permutation?? Answered by Penny Nom. 





A 4 digit phone lock using the digits 4, 2, 3 and 0 
20121224 

From Ragu: I would like to know formula for searching a combination of 4 digit. eg.
I knew my Phone Lock combination is my car plate number 4230. But
this time, i used another way around instead of 4230.
But I am sure, there is only this 4 digits only. 4 & 2 & 3 & 0 Answered by Penny Nom. 





A security code 
20120901 

From Breanna: The code is made up of five two digit numbers from 01 to 20 entered in a certain order. How many different security codes are possible with this system Answered by Penny Nom. 





6/49 
20120525 

From Mark: Can you explain why the lotto 6/49 (a lottery of 49 numbers and 6 slots) is a combination and not a permutation?
It seems to me that it would a permutation given that the numbers are written in numeric "order" when you look them up in the newspaper.
I realize that 49P6 is orders of magnitude larger than 49C6. But I am confused about the reasoning behind it. Answered by Robert Dawson. 





A lost combination 
20120306 

From pam: I lost the combination for my luggage that I haven't used for many years. It is a 3 digit combination. I'm pretty sure only these numbers could have been used: 0,1,2,3,7,8,9. Can you give me a calculated list of all these combinations? Answered by Penny Nom. 





In how many ways can 12 members of volleyball team line up ... 
20111118 

From Aatir: In how many ways can 12 members of volleyball team line up if the captain and assistant captain have to remain together. Answered by Penny Nom and Stephen La Rocque. 





Arrangements of ten letters 
20110415 

From Mustafa: In how many ways can the ten letters of the word GELATINOUS be arranged
in a line so that the vowels are in alphabetical order (not necessarily together)
when read from left to right? Answered by Penny Nom. 





Is this a permutation or a combination? 
20110111 

From Lamont: If I have 10 marbles: 3 blue, 2 yellow, 1 purple, and 4 red
How many ways can I pull out 3 different sets of marbles?
Is this a permutation or a combination? Answered by Penny Nom. 





Six numbers from 1 to 20 
20100510 

From martine: how many different combinations can be made from numbers 1 to 20 using them only once in a six number scale Answered by Penny Nom. 





6 items are taken 3 at a time 
20100412 

From Kristen: Explain the three different scenarios that could occur and would result in the following number of outcomes when 6 items are taken 3 at a time. Give reallife examples.
a. 63 = 6 x 6 x 6 = 216 possibilities
b. 6! / (6  3)! = 6 x 5 x 4 = 120 possibilities
c. 6! / (6  3)! 3! = 20 possibilities Answered by Tyler Wood. 





A student wants to take four courses 
20100221 

From deciree: A student wants to take four courses. She has 11 courses to choose from: 6 sophomore level and 5 freshmen level.
Each course is available at 8,9,10,11 am.
a)if the student wants to take 2 freshmen and 2 sophomore courses, how many choices she have?
b)If she wants to alternate these courses, freshmen,sophomore,freshmen,sophomore, how many choices does she have? Answered by Lorraine Dame. 





Two digit numbers 
20091210 

From Jim: Can you tell me how many 2 digit numbers can be created
from the following:
1118539 Answered by Penny Nom. 





Airline routes 
20091201 

From Kapilan: Please answer the following question:
From A to B there are four possible air routes. From B to C there are five
possible air routes. From C to D there are three possible air routes. How many
different trips can be taken from A to D and back without taking the same route on any
section of the return trip? Answered by Penny Nom. 





Sixletter words 
20091124 

From christine: How many sixletter words (not necessarily an English words) are there in which exactly three of the letters are z's? Answered by Claude Tardif. 





6 numbers from 12 
20091114 

From david: how many combinations of 6 numbers can there be using 12 different
numbers without using the same numbers more than once?
2, 3, 5, 8, 9, 11, 12, 14, 17, 19, 20 & 31. Answered by Penny Nom. 





4 couples seating themselves at a round table 
20090904 

From Hui: Q: 4 couples seating themselves at a round table. Men must seat together and women seat together. How many ways are there? Answer in the printed text is 576. I got 144. Pls advice. Answered by Claude Tardif. 





A three digit number 
20090904 

From Mark: How many 3 digit positive integers are odd and do not contain the digit 5? Answered by Penny Nom. 





Arrangements 
20090820 

From Anna: How many arrangements of FATHER can be made if all the vowels must be kept together? Answered by Leeanne Boehm. 





Rearrangements of 8028842 
20090717 

From Con: How many different 7digit numbers can be made by rearranging the digits in the number 8028842? Answered by Penny Nom. 





Passwords 
20090708 

From Tim: pretend I have a password that is  9k7d4112p5o
assuming that you continuously guessed wrong until that exact
combination was the only possible answer left;
how many times would you have to guess?
capital letters are not relevant
and you do not know how many digits there are Answered by Stephen La Rocque. 





All permutations of 1  5 
20090603 

From chris: i need a list of 5 digit number combos using numbers 15 Answered by Penny Nom. 





Combinations and permutations 
20090507 

From Renee: We are trying to figure out the equation or how to solve the problem of how many combinations of 4 digits you can get out of 09 without repeating the numbers in a set (i.e., no 0001, 0002) begining with 0123, 0124, 0125 etc
Thanks! Answered by Stephen La Rocque. 





Permutations 
20090430 

From Evan: There are 24 possible combinations of the numbers 1 through 4, and 5 times that many
combinations of the numbers 1 through 5. Are there 144 (6x24) possible combinations of 1 through
6? Or is it 6x5x24? Is there a formula for determining this?
Thanks,
Evan Answered by Penny Nom. 





Possible combinations of 2 numbers between the range 1 and 50 
20090403 

From Carl: Hi, I think I'm right in saying that there are 1225 possible combinations of 2 numbers between the range 1 and 50?
I used the function =combin(50,2) in excel to establish this.
What I'd like to do is be able to prove this using a calculator or pen and paper.
I've searched your website, but to be honest all the algebra stuff just confuses me!
Could you explain the math without using algebra please.
(We run a tote in the office, where you pick any two numbers between 1 and 50, and if your numbers come up you win, so I'm trying to work out all the possible combinations. The same number can't be used twice either)
many thanks
Carl Answered by Stephen La Rocque. 





Ranking 6 players from 49 
20090324 

From liz: My daughter is having problems with this question,
if you had to match 6 players in the correct order for most popular outfielder from a pool of professional players numbered 1 through 49 , how many possibilities are there? Answered by Penny Nom. 





Trifecta 
20090309 

From Annie: I want to know how many combinations are there for the first 3 horses in
a race of 20 horses. For example, the horses can come in 1, 5, 7 or
7, 5, 1 which are totally different combos. I've only found formulas
where the 1, 2, 3 and 3, 2, 1 are counted as the same combo.
In horse racing every combo is different.
Thank you. Answered by Penny Nom. 





4 digit strings starting with 7 
20090303 

From Angel: I need to no all the 4 digit combinations starting with 7 and with no repititions. Answered by Penny Nom. 





Three number combinations 
20090227 

From mark: How many three number combinations can I get using 1,2,3,4,5? I can use a number more than once. Answered by Robert Dawson. 





Some 9 letter permutations 
20090224 

From Cecilia: If the letters G, H, S are used exactly 3 times how many 9 letter combinations can be generated? Answered by Penny Nom. 





Seating 8 people in a row 
20090215 

From raymond: In how many ways can 4 boys and 4 girls be seated in a row if they may sit anywhere Answered by Penny Nom. 





Seating around a circular table 
20090209 

From Jenny: A committee of 15  9 women and 6 men  is to be seated at a circular table (with 15 seats). In how many ways can the seats be assigned so that no two men are seated next to each other? Answered by Victoria West. 





Four digit numbers using 1, 0, 2 and 8 
20090127 

From Mike: How many combinations of this four digit number 1028 can this produce only using those numbers. Example would be 0000, 0001,0002,0008,0010,0011,0012,0018,etc... Answered by Penny Nom. 





5 "digit" combinations 
20090122 

From Timothy: I was wondering how many different 5 digit combinations are possible using 09 and az and you can use the same letter or number repeatedly in the combination. Answered by Penny Nom. 





Permutations and combinations 
20081228 

From KHADIJA: What's the difference between Permutation and Combination? Answered by Penny Nom. 





Choosing 5 numbers from 39 
20080804 

From robert: Hi,
I am sending you the web page where I found the question of 39 numbers and =
5 choices, please see below.
How does they comes up with the 120 combinations?
How can I comes up with the 120 combinations in a printout form?
Please read all the way to the bottom of there example and see where they talks about the 120 combinations. Answered by Victoria West and Harley Weston. 





How many different ways can cards fall when shuffling? 
20080609 

From Twyla: How many different ways can cards fall when shuffling? In other words,
if you have 52 numbers, each one unique, how many different ways can
they be arranged, using each number only once? Answered by Penny Nom. 





7, 6, 5, 4, 3 and 2 digit numbers 
20080210 

From Edward: using the number 12345678 how many 7, 6,5,4,3 & 2 digit numbers can you make when not using any number twice in the same number Answered by Penny Nom. 





Permuting the letters in ELEVEN 
20080123 

From Beth: (i) Find the number of distinct permutations that can be formed from all
of the letters of the word ELEVEN.
(ii) How many of them begin and end
with E?
(iii) How many of them have the 3 E's together?
(iv) How many
begin with E and end with N? Answered by Stephen La Rocque. 





Five boys and two girls in a line 
20080112 

From pankaj: There are 5 boys and 2 girls.In how many ways can they stand in row so that no two girls are together? Answered by Penny Nom. 





Rearranging the alphabet 
20070818 

From samhita: Find the number of ways of arranging 26 letters in the English alphabet in a row such that there are exactly 5 letters between x and y. Answered by Stephen La Rocque and Penny Nom. 





How many different orders are there? 
20070813 

From Ethel: This is a question from the PTSP Success Test. There are 4 horses at a stable. And how many different orders can they be ridden by their owners? The answer choices are A. 4 B. 10 C. 16 D. 24 and E. 36 Answered by Harley Weston. 





Permutations and combinations 
20070729 

From Marcy: I'm having difficulty teaching my students when to use permutations or combinations. Is there a "catchy" way to teach them that will help them to identify when to use permutations and combinations? Answered by Penny Nom. 





Five button push lock 
20070706 

From Jeffrey: We have a 5 button push lock and we were wondering how many different codes were possible in this situation. In this lock each button can only be pressed once (ie it's not a numbered key pad) and order matters. How would we find out the # of possible codes? Answered by Stephen La Rocque. 





A push button lock 
20070521 

From Werner: I have a push button lock (the numbers entered does not have a start and end i.e. 1234 = 4321 or 2341 = 3241 ...)
So if it is a 4 digit code and you have 10 buttons (0 to 9) there should be 5040 (?) possibilities and how would I list these unique combinations Answered by Penny Nom. 





All the possible 8 digit combinations of 09 
20070516 

From Christofer: What are all the possible 8 digit combinations of 09? List please
I know that there are 43046721 combinations but I want to know
what they are. Answered by Stephen La Rocque. 





4 digit lock 
20070516 

From Kathy: Hi,
My name is Kathy and I'm a REALTOR, I have 2 different lockboxes that
I cannot remember the combinations for, so what I need is all the
different combinations for 4 digits that cannot be the same, i.e. 0123.
Can you help me out? I would hate to throw them away as they cost
about $30 each.
Thank you,
Kathy Answered by Stephen La Rocque. 





How many ways can I choose 4 numbers from 10? 
20070510 

From Mitch: Can you give me a list of all possible four digit combinations using 0 to 9 no repeat numbers and the numbers do not have to be entered in a special order. If you could also show me how to calulate this it would be great Answered by Stephen La Rocque. 





Permutations, probability and standard normals 
20070509 

From Katrina: I have a few problems that i seem to be stuck on or can not start. Can you please help me ?
1) There are 20 people on an event planning committee. How many differnt ways can a
chairperson and assistant to the chairperson be selected?
2) An unprepared student makes random guesses for 10 true or false questions on a quiz.
Find the probability that the student passes the quiz by guessing 7 of the questions
correctly.
3) The heights of 18 year old men are normally distributed with a mean of 68 inches and
a standard deviation of 3 inches. If a random sample of 25 18year old men is selected
what is the probability that the mean height is between 68.5 and 72 inches? Answered by Penny Nom. 





Four people in a car 
20070324 

From shannon: If there are 4 people in a car and only 1 can drive how many possible ways the the people sit in the car? Answered by Penny Nom. 





Permutations 
20070311 

From Meghan: Hello, I'm working on the beginning of the combinatorics unit in Math 12 in BC. We just looked at using the nPr or n!/(nr)! formula.
My questions for homework that I didn't quite understand how to do were:
Solve each equation for n. 1) nP2 = 90 2) 6Pn = 720
I managed to do these in my head, but I'm not sure how to solve these algebraically.
Thanks for all your help! Meghan Answered by Penny Nom. 





How many sequences are possible? 
20061028 

From Sam: A fair eightfaced die with faces numbered 1,2,3,4,5,6,7 and 8 is tossed six times and the sequence of numbers is recorded. How many sequences are possible? Answered by Penny Nom. 





I have 4 numbers and want to find all possible combinations. 
20060905 

From Candace: I have 4 numbers and want to find all possible combinations. I come up with 24 but that seems too small...am I correct that there should be more? Answered by Stephen La Rocque. 





Bike combination lock 
20060716 

From Alejandra: I have a bike lock. It has three columns of the digets 19 and zero. I have forgotten the combination. Three numbers need to be imputed. What are the possible lock cominations, so that i can try to break the lock? Answered by Stephen La Rocque. 





A 5 digit, pushbutton door 
20060620 

From George: We have a 5 digit, pushbutton door look (15). How many combinations are possible without repeating a number in a 3 digit entry code? Answered by Stephen La Rocque. 





How many arrangements are there? 
20060416 

From Jim: Can you tell me how many different combinations there are using 3 letters and 4 numbers. Letters first then numbers, example aaa0001, aaa0002, aaa0003,baa0001, baa0002, zzz9999. Answered by Stephen La Rocque. 





How to find the odds for a lottery jackpot 
20060412 

From Harvey: Is it best to use factorials to calculate the odds of winning a lottery, such as the MegaMillions that is popular in the US, or is there a better way? Answered by Stephen La Rocque. 





Permutations and probability 
20060327 

From Alyssa: I am a high school student, my teacher gave the class a worksheet on permutations and probability and told us to do independent work when we have not covered the material yet and he will not answer any questions. I am lost and don’t know where to begin. Can you help?
1. Find the number of 6letter permutations of all the letters in EUCLID that end with either the letter E or the letter D?
Aluminum chips A, B. C, and D weigh 1g. 5g. 10g. and 20g. respectively. How many different masses can be measured by using one or more of the 4 weighs on a balance scale? Answered by Stephen La Rocque. 





Four digit numbers 
20060325 

From Kenvin: i was wondering how many possibilities is there for a four digit number with the same first two numbers and the last two is not one of the first with numbers 19 Answered by Stephen La Rocque. 





Five digit numbers 
20060325 

From Tony: Can someone tell me how many unique five digit numbers can be generated using 09? Answered by Stephen La Rocque and Penny Nom. 





A 4digit Combination Lock 
20060309 

From Adam: I have a 4digit Combination Lock but i have forgotten the code. i do know that it is an even number, all the digits are different, the second digit is zero and the largest digit is 6. what are all the possibilities. Answered by Stephen La Rocque. 





A combination lock 
20051003 

From A bicycle owner: What are the number of permutations (Combinations?) of a combination lock of a cylinder with four separate movable rings. Each ring has the numbers from one to six (six "permutations" per ring).
This is a bicycle lock.
I was told that it is 6^{2} (6x6x6x6) = 1296.
Is this correct?.
Answered by Penny Nom. 





In how many ways can 12 different fruits be distributed 
20050613 

From Sapta: In how many ways can 12 different fruits be distributed among 3 boys so that each receives at least 1 fruit? Answered by Claude Tardif. 





A six digit answer 
20050518 

From Angela: I have a 6 digit answer that is comprised of only 4 symbols: a square, a circle, a triangle, and an X. How many possible solutions are there and how is the best way to find out what ALL the solutions are? Answered by Penny Nom. 





How many three digit numbers can I make from the four cards 
20050312 

From Sue: I have four playing cards 2,3,4,5 of hearts. How many three digit numbers can I make from the four cards. Answered by Penny Nom. 





A seating arrangement 
20050203 

From Kay: If you have 5 married couples, how many arrangements can be made if they must sit across from each other? Answered by Penny Nom. 





Visitor wins or home wins 
20040929 

From Paavn and Laddi: I have a math question i am stumped on for one of my classes. The question states that there are 14 baseball games with 2 possible outcomes for each game, visitor win or home win. How many different total combinations are possible from the 14 games? Answered by Penny Nom. 





Why the word combinations? 
20040505 

From Megan: Why are locker combinations called locker combinations not locker permutations. The order of the numbers matter so why aren't they called locker permutations. Answered by Claude Tardif and Penny Nom. 





Combinations of 12 numbers 
20040118 

From Cheryl: I would like to know the total possible combinations using the numbers 112. We would also like to know the formula used to calculate the number of combinations. Answered by Harley Weston. 





Permutations and combinations 
20040111 

From A student: Are there more permutations than combinations for a series of 4digit numbers with the integers 09? Can you illustrate calculations? Answered by Penny Nom. 





Arrangements around a circular table 
20031007 

From Lori: How many arrangements can 6 people be seated round a circular table? Answered by Penny Nom. 





Mary typed a six digit number 
20030306 

From Pillar: Mary typed a six digit number, but the two 1's did not show. What appeared instead was 2002. How many different sixdigit numbers could she have typed? Answered by Penny Nom. 





The number of possible outcomes with 8 games 
20030221 

From Gianni: If 8 basketball games being played(no ties), which means a total of 16 teams, what are the total number of possible outcomes that can occur. Answered by Penny Nom. 





Combinations of 1,2,3,4,5 and 6 
20030201 

From Kyle: Today I was aked a question by my geometry teacher. It was, what is the sum of all the possible combinations of the numbers 1, 2, 3, 4, 5, and 6. He said all the possible combinations were 720. I would like to know how u could do this, and what the answer is. Answered by Penny Nom. 





A penny, a nickel, a dime, a quarter, and a loonie 
20030121 

From Blaine: I have 5 coins: a penny, a nickel, a dime, a quarter, and a loonie. How many different amounts of money could I pay using any combination of these coins? Answered by Penny Nom. 





ALL of the possible 3 digit combinations of 0  9 
20021209 

From Melissa: I need to find out ALL of the possible 3 digit combinations of 0  9. Eg: 000,001,002,003. This is going to take me VERY LONG time. Any suggestions. Answered by Penny Nom. 





The horse race 
20021119 

From Pat: The Horse Race Three horses run a race. In how many different ways can the three horses finish the race if ties are allowed? Answered by Penny Nom. 





How many empty seats are there? 
20021029 

From Ronnie: Three new students arrived in class. Teacher says there are 210 ways you can set. How many empty seats are there in the classroom? After new student are assigned seats, how many empty seats will remain? Answered by Penny Nom. 





Designing a ballot 
20020426 

From Kelley: I want to design a ballot for four elections. Actually all the candidate races are on 1 ballot. I need to know how many different ballot styles would be needed for all of the candidates to be in each rotation an equal number of times. For example:  A,B,C & D are running for mayor
 E,F,G & H are running for congress
 I,J & K are running for senate
 L,M & N are running for governor
They are all on the same ballot. But in each race their name (for instance A) has to be in the #1 rotation, #2 rotation, #3 rotation, and #4 rotation for his race on this ballot an equal number of times as B,C and D. The same goes for the other candidates for their perspective races. How many total ballot styles will there be? Answered by Penny Nom. 





How many different account numbers are possible? 
20020418 

From Andrene: how many different account numbers are possible if the account numbers consist of a letter of the alphabet, followed by five numerical digits, and followed by another letter. Answered by Penny Nom. 





Combinations? 
20020316 

From Sara: A combination lock has three dials numbered 0  9. Give a mathematical reason why "combination" is a bad name for this type of lock. Answered by Penny Nom and Chris Fisher. 





Permutations and combinations 
20011203 

From Linda: There are 12 people who can be chosen for a project. Q1. Suppose that 2 people refuse to work together. How many groups of 7 can be chosen to work on a project? Q2. Suppose that 2 people insist on working together or neither will work on the project. How many groups of 7 can be chosen to work on the project? Answered by Penny Nom. 





How many 6 digit numeral combinations 
20010426 

From Bryan: How many 6 digit numeral combinations are there If you can only use 6 digits (1, 2, 3, 4, 5, 6) and no number can be used twice in the Numeral! Answered by Claude Tardif. 





A combination lock 
20010405 

From Ken: What is total number of combinations possible on a combination lock which contains 3 wheels each numbered 09? Answered by Penny Nom. 





Four crayons 
20010110 

From Neyra Espinoza: You have four crayons (red, blue, yellow, green). If you line them up, how many different combinations can you get? Answered by Patrick Maidorn. 





Permutations vs. combinations 
20001120 

From Ashleigh: How do you tell whether the problem is a permutation or a combination? Answered by Walter Whiteley. 





Permutations and Multiplication Principle 
20000922 

From Candice: A forester selects 4 pink and 4 white dogwoods. The trees are to be planted in row. If a tree is distinguished by color only, in how many ways can the eight dogwoods be planted? How many of these arrangements have at least two trees of the same color side by side? Answered by Denis Hanson and Claude Tardif. 





Circular Permutations 
20000918 

From J. Stuart: I am able to understand the simple theory behind circular permutations (that is, there are (n1)! ways to arrange n people in a circle. One question, however, is giving me some difficulty. It reads: In how many ways can four married couples be arranged around a circular table if each man must sit beside his wife? Any suggestions you have would be greatly appreaciated. Answered by Penny Nom. 





Question regarding Combinations and permutations 
20000419 

From Paul Stoner: I am a programmer and I have a quick question. I have 5 criteria which will either be true or false. If a criterion is true it returns its ordinal representation, if false it returns 0. I then combine these results to determine where to go in my program. My question is: How can I figure out the number of combinations I will need? Answered by Patrick Maidorn. 





Ten stools 
20000206 

From Haim: In a cafe with ten stools, three customers want to be seated, so that no two are next to each other. How many ways can this be done? Do not consider it to be a separate seating if two customers switch seats. Answered by Denis Hanson and Walter Whiteley. 





Trolls and Gargoyles 
19991102 

From TexGrimm: How can you seat 6 monsters  3 Trolls and 3 Gargoyles at a circular table if the trolls look alike and the gargoyles look alike? Does your formula work for 9 monsters  4 trolls and 5 gargoyles? Answered by Penny Nom. 





Circular Permutations 
19970205 

From Ed Varjassy: I have an understanding of simple problems involving circular permutations but do not understand them when they start to have restrictions. Is there any advice you can give for these more complicated permutations. Answered by Penny Nom. 





Divisibility of 2n choose n. 
19960924 

From Kathy Doan: Can you prove that "2n choose n" is not divisible by 3, 5, and 7 for infinitely many n? Answered by Penny Nom. 





How many possible combinations of answers are there? 
19960920 

From David Bester: Suppose there is a 12 question test. Each question can be answered only by true or false. Assuming that all questions are answered (i.e. none are left blank), how many possible combinations of answers are there? And what formula was used to arrive at the answer? Answered by Denis Hanson. 

