(a) We automatically put Corey and Peter on the team, so we have 3 choices left to make with 8 people to choose from, so the answer is 8C3 = 56.
(b) We want to count these cases seperately. With just Corey we have 8C4 = 70 ways to choose a team and the same goes for Peter. We add 70+70+56 = 196 lineups.
Well, none of the digits can be 0 because the product is than going to be 0. You can't have more than 2 1's either.
To get started, we turn to the factorization of 10,000. It factors out as 2^4 * 5^4. This means that the only possible combinations of numbers are:
1,2,8,5,5,5,5 -> 105 ways to arrange this
2,2,4,5,5,5,5 -> 105 ways to arrange this
1,4,4,5,5,5,5 -> 105 ways to arrange this
1,1,8,5,5,5,5 -> 105 wasy to arrange this
So the number of different possible codes is 105+105+105+105 = 420 possible codes.
First, do you know what palindromes are? Palindromes are numbers that are the exact same when read from right to left or left to right. For example, 1001 is a palindrome since it reads 1001 from right to left and it also reads 1001 from left to right.
Another hint: Think about how many digits those numbers from 1000 - 100000 have. Could you possibly think of a way to use that?