If n is a positive integer how many 4tuples of integers from

If n is a positive integer, how many 4-tuples of integers from 1 through n can be formed in which the elements 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 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

Solution

5. In this question where the numbers might not be necessarily distinct,

   Selecting 4 numbers in the ascending order from 6 numbers,

   = (n+r-1)! / r! (n+r-1-r)!

    = (6+4-1)! / 4! (6+4-1-4)!

    = 9! / 4! 5!

    = 126

Therefore in 126 ways they can be arranged

6. (7+5-1)! / 5! (7+5-1-5)!

    = 11! / 5! 6!

    = 462

Therefore in 462 ways they can be arranged