What is combination with replacement?

Combination with replacement is the number of combinations when \(r\) objects are retrieved with duplicates from \( n\) different ones.

Combination with replacement is defined by the following formula

\[ \mathrm{C}^{\mathrm{R}}\left(n,r\right)=\frac{\left(n+r-1\right)!}{r!\left(n-1\right)!} \]

Example

Example 1

Calculate \( \mathrm{C}^{\mathrm{R}}\left(4,3\right) \).

Solution

\begin{align} \mathrm{C}^{\mathrm{R}}\left(4,3\right)&=\frac{\left(4+3-1\right)!}{3!\left(4-1\right)!}\notag\\[+10pt] &=\frac{6!}{3!\cdot3!}\notag\\[+10pt] &=20\notag \end{align}

Example 2

Calculate \( \mathrm{C}^{\mathrm{R}}\left(2,3\right) \).

Solution

\begin{align} \mathrm{C}^{\mathrm{R}}\left(2,3\right)&=\frac{\left(2+3-1\right)!}{3!\left(2-1\right)!}\notag\\[+10pt] &=\frac{4!}{3!\cdot1!}\notag\\[+10pt] &=4\notag \end{align}