Question

How many 4-permutations of [10] have maximum element equal to 6? How many have maximum element at most 6?

Answer 1

I'm guessing that [10] refers to the set of the first 10 positive integers.

If the largest element of a given 4-permutation is 6, then the other three elements are pulled from the set {1, 2, 3, 4, 5}. This can be done in 5!/(5 - 3)! = 60 ways. Then there are four possible positions to place the 6, giving a total of 4 * 60 = 240 permutations.

If the largest element of a permutation is *at most* 6, then the maximal element is 4, 5, or 6.

• If it's 4, then there are three other elements available; this can be done in 3!/(3 - 3)! = 6 ways; multiply by 4 to get a total of 24;
• If it's 5, then there are four other elements available, hence 4!/(4 - 3)! = 24 ways; multiply by 4 to get a total of 96;
• If it's 6, then the total is 240.

Putting everything together, the total number of permutations in which the maximal element is at most 6 is 24 + 96 + 240 = 360.