S = {1, 2, 3, 4, 5 … N}, where N is greater than 5. X is a set consisting of all the possible four-element subsets of S. The sum of all the numbers in all the elements of X is ______.

The problem is asking for the sum of all numbers in all possible four-element subsets of the set S.

Here's how to solve it:

Step 1: Understand the problem The set S = {1, 2, 3, 4, 5 … N}, where N > 5. The set X consists of all possible four-element subsets of S. We need to find the sum of all numbers in all elements of X.

Step 2: Find the total number of four-element subsets The number of ways to choose 4 elements from a set of N elements (also known as "N choose 4") is given by the combination formula: C(N, 4) = N! / [(N-4)! * 4!]

Step 3: Find the sum of all elements in S The sum of all elements in S is given by the formula for the sum of an arithmetic series: Sum = N*(N+1)/2

Step 4: Find the sum of all numbers in all elements of X Each number from 1 to N will appear in C(N-1, 3) subsets of X (this is because for each number, we are choosing the remaining 3 elements from the remaining N-1 numbers). So, the sum of all numbers in all elements of X is given by: Sum * C(N-1, 3)

Step 5: Simplify the expression We can simplify the expression by substituting the formulas from steps 2 and 3 into step 4. This gives us: [N*(N+1)/2] * [(N-1)! / [(N-4)! * 3!]]

This is the final answer.