If n is a positive integer, how many 5-tuples of integers from 1 through n can be formed in which the elements of the 5-tuple are written in decreasing order but are not necessarily distinct? In other words, how many 5-tuples of integers are there with ?
To find how many tuples of integers are there with .
Any quintuple , with can be represented as a string of vertical bars and crosses. The positions of the crosses indicate which integers from are indicated in the tuple.
The number of combinations with repetition allowed that can be selected from a set of elements is .
Thus, the number of such quintuple is the same as the number of strings of vertical bars and crosses.
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