Quandaries and Queries


My level of question is secondary.
 How many different combinations can be made of the numbers 1-7 in 4 string combinations in any order? EX:7-2-3-5, 3-5-1-7, etc...
 And if you were to include the same 2 numbers in every combination, how many would that make? EX: Using 1 & 2: 1-2-7-4, 5-1-4-2, 2-7-5-1, etc...
 Thank you for your time and effort.


Hi Tom,

In your first problem the examples you list don't have any digits repeated so I am going to assume that you are looking for 4 digit strings made from the digits from 1 to 7 with no digit repeated. Think about constructing such a string. You start by writing one digit. You have 7 choices of digit. Now you write down a second digit. This time you only have 6 choices since you can't repeat the first digit. That is each one digit number you can extend to a two digit number in 6 ways.

Thus there are 7 6 ways to construct a two digit number.

Now add a third digit, you have 5 choices. That is each two digit number you can extend to a three digit number in 5 ways.

Thus there are 7 6 5 ways to construct a three digit number.

Likewise for the fourth digit.

Thus there are 7 6 5 4 ways to construct a four digit number.

For your second problem I am going to assume that the two special digits are known before you start, so in your example you know that 1 and 2 must be included. Place the two special digits first. You have 4 choices for the position of the first special digit and then 3 choices for the position of the second.

Thus there are 4 3 ways to place the two special digits.

Now you have two more positions to fill from the five remaining "non-special" digits. Chose one of them and put it in the first vacant position. You have 5 choices as to which digit you chose. Now chose one of the four remaining to go into the one remaining position. There are 4 choices.

Thus there are 4 3 5 4 combinations.



Go to Math Central