$$\binom{n}{k}=\frac{n!}{k!(n-k)!}$$ The expression above is also known as binomial coefficient. Is there a similar naming convention that describes the one below? $$k!\binom{n}{k}=\frac{n!}{(n-k)!}$$
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4$\begingroup$ It's sometimes referred to as $P^n_k$, the permutation of $k$ objects from a given group of $n$ $\endgroup$– user418131Sep 17, 2018 at 13:22
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2 Answers
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$\displaystyle \frac{n!}{(n-k)!} = k!\binom{n}{k} = n(n-1)(n-2)\cdots (n-k+1)$ is the falling factorial.
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2$\begingroup$ It is sometimes denoted $n^{\underline k}$ to illustrate the parallels with exponentiation (instead of multiplying k copies of n, it is k shifted copies) $\endgroup$ Sep 17, 2018 at 13:50
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If the choose function is defined as: $$C(n,r)=\frac{n!}{r!(n-r)!}$$ And the permutation function is defined as: $$P(n,r)=\frac{n!}{(n-r)!}$$ then you are looking at the permutation function
answered Sep 17, 2018 at 14:54