Prvoe the number of k-compositions of n is n−1 k−1

Question

Prvoe the number of k-compositions of n is n−1 k−1

Answer 1

Reference: https://proofwiki.org/wiki/Number_of_Compositions#:~:text=A%20k%2Dcomposition%20of%20a,i%20are%20strictly%20positive%20integers.

As defined in the proof wiki page, suppose we have a list of n 1's separated by n-1 blanks which can either be a comma or a plus sign. If we have n-1 total blanks, and we wish to count all possible lists of length k (i.e. k-1 separations), then this corresponds exactly to the number of ways of choosing k-1 blanks to be commas out of n-1 blanks total. This number is (n-1) choose (k-1) by definition (i.e. from the binomial theorem).