Question

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

Answer 1

The number of 4-permutations of [10] with a maximum element of 6 is 24. The number of 4-permutations of [10] with a maximum element at most 6 is 120.

Step-by-step explanation:

To find the number of 4-permutations of [10] with a maximum element of 6, we can consider the possibilities for the position of the maximum element in the permutation. There are 4 possible positions for the maximum element: first, second, third, or fourth. If the maximum element is in the first position, the remaining 3 elements can be any combination of the remaining 3 numbers (10, 9, and 8) which gives us 3! = 6 permutations. Similarly, if the maximum element is in the second, third, or fourth position, we will have 6 permutations for each position.

Therefore, the total number of 4-permutations of [10] with a maximum element of 6 is 4 * 6 = 24.

To find the number of 4-permutations of [10] with a maximum element at most 6, we need to consider all possible values for the maximum element. We already found that there are 24 permutations with a maximum element of 6. Now, we need to consider the possibilities where the maximum element is 5, 4, 3, 2, or 1.

If the maximum element is 5, the remaining 3 elements can be any combination of the remaining 4 numbers (10, 9, 8, and 6) which gives us 4! = 24 permutations. Similarly, if the maximum element is 4, 3, 2, or 1, we will have 24 permutations for each maximum element.

Therefore, the total number of 4-permutations of [10] with a maximum element at most 6 is 24 + 24 + 24 + 24 + 24 = 120.

Answer 2

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.