File:Combinations with repetition; 5 multichoose 3.svg

There's a bijection between

• the k-element multisets with elements from an n-element set    (k-combinations of n elements with repetitions)
• and the k-element subsets of an n+k−1-element set    (k-combinations of n+k−1 elements without repetitions).

${\displaystyle \left(\!\!{n \choose k}\!\!\right)={\binom {n+k-1}{k}}}$

This file shows the bijection between

• the 3-element multisets with elements from ${\displaystyle \{1,2,3,4,5\}}$    (right side)
• and the 3-element subsets of a 7-element set    (left side).

${\displaystyle \left(\!\!{5 \choose 3}\!\!\right)={\binom {5+3-1}{3}}={\binom {7}{3}}=35}$

The first line represents the following correspondence between a 7-tuple, a 5-tuple and a 3-element multiset:

(1,1,1,0,0,0,0) ≙ (3,0,0,0,0) ≙ {1,1,1}