Sender: katie Hi! My name is Katie. I am in seventh grade. I (being a student) wish to know the answer to this question.. Okay.......here's my question...how many different ways can you form a four digit number out of these digits..1,2,3,4,5,6,7,8? This is how my teacher said to do this.... 8*8, 7*7,6*6,5*5,4*4,3*3,2*2,1*1. then she said to add up the products and to multiply by 7. is this correct, and if not how can you figure this out.
thanks sooo much, The answer your teacher gave you only adds up to 1428 ways but we can clearly have more than that. Think of the problem this way  suppose that you are to make a 4 digit number using digits from the set {1,2,3,4,5,6,7,8} and we also insist that you do not repeat a digit. Think of this as filling in a digit in each of the 4 spots _ _ _ _. There are 8 ways to choose a digit to place in the first spot. Now we have to fill in a digit in each of the remaining 3 spots X _ _ _. There are 7 ways to choose a digit to place in the 2nd spot (remember, we've used one digit already and can't repeat it). Now we have to fill in a digit in each of the remaining 2 spots X X _ _. There are 6 ways to choose a digit to place in the 3rd spot (remember, we've used two digits already and can't repeat them). Now we have to fill in a digit in the remaining spot X X X _. There are 5 ways to choose a digit to place in this last spot (remember, we've used three digits already and can't repeat them). Thus, all told the number of ways is 8x7x6x5 = 1680 ways to do this. What happens if we allow repetitions of digits  that is, allow numbers such as 3539 or 1121? We haven't counted these yet so there are actually more than 1680 ways to form 4 digit numbers if we allow repetitions too. You should ask your teacher if repetitions of digits is allowed. It actually becomes easier to answer in that case as we have 8 choices for the 1st spot, 8 choices for the 2nd spot, 8 choices for the 3rd spot, 8 choices for the 4th spot, giving us 8x8x8x8 = 4096 ways to make a 4 digit number from {1,2,3,4,5,6,7,8}. If this is confusing try some simpler question. In how many different ways can you form a two digit number out of these digits..1,2? Answer: Let's list them up and count them:
Using the argument above, we have 2 choices for the 1st spot and 2 choices for the 2nd spot giving us 2x2 = 4. In how many different ways can you form a three digit number out of these digits..1,2 ? Answer: Let's list them up (in order) and count them: 111, 112, 121, 122, 211, 212, 221, 222. There are eight ways to form such a number. Using the argument above, we have 2 choices for the 1st spot, 2 choices for the 2nd spot and 2 choices for the 3rd spot giving us 2x2 = 4. What if we are allowed to use the digits {0,1,2,3,4,5,6,7,8,9}? We have to be careful here as we do not usually allow 0057 as a 4 digit number, we simply write 57.
Cheers,
