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Combinatorics

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Compositions As Binomial Coefficient

COMBIN-3CJJ1G

Let $n$ be a positive integer. A composition of $n$ into $k$ parts is given by $n = m_1 + \ldots + m_k$ where each $m_i > 0$.

How many compositions of $n$ into $k$ parts are there? (Note: $3 = 1+2$ and $3 = 2+1$ are different compositions of $3$ into $2$ parts.)

A

${n \choose k}$

B

${n \choose k-1}$

C

${n - 1 \choose k - 1}$

D

None of the above