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 ?
To find how many tuples of integers are there with .
Any quadruple with , can be represented as a strings of vertical bars and crosses.
The number of combinations wflh repetition allowed that can be selected from a set of elements is .
Any quadruple with , can be represented as a strings of vertical bars and crosses. The positions of the crosses indicate which integers from are included in the tuple.
Thus the number of such quadruples is the same as the number os strings of vertical bars and crosses.
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