If n is a positive integer, how many 4-tuples of integers from 1 through n can be formed in which the element of the 4-tuple are written in increasing order but are not necessarily distinct? In other words, how many 4-tuples of integers (i, j, k, m) are there with 1 ≤ i ≤ j ≤ k ≤ m ≤ n ?

To find how many 4 tuples of integers {i,j,k,m} are there with 1≤i≤j≤k≤m≤n.

Given information:

Any quadruple {i,j,k,m} with 1≤i≤j≤k≤m≤n, can be represented as a strings of n−1 vertical bars and 4 crosses.

Concept used:

The number of r combinations wflh repetition allowed that can be selected from a set of n elements is (r+n−1r).

Calculation:

Any quadruple {i,j,k,m} with 1≤i≤j≤k≤m≤n, can be represented as a strings of n−1 vertical bars and 4 crosses. The positions of the crosses indicate which 4 integers from 1 to n are included in the n tuple.

Thus the number of such quadruples is the same as the number os strings of (n−1) vertical bars and 4 crosses.

i.e