In how many ways can we place $k$ distinct objects into $n$ identical boxes if each box contains atmost one object

Question

In how many ways can we place $k$ distinct objects into $n$ identical boxes if each box contains at most one object?

Interpreting it as permuting n objects taken k at a time.So,there are $\binom{k}{n}\times n!$ ways.

Am i correct?

Answer 1

Informally:

Each object goes into a box by itself. This can only be done one way if $k\leq n$ and zero ways if $k>n.$

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More Formally:

We are counting the number of partitions of $\{1,2,\ldots,k\}$ with at most $n$ parts and each part has at most one element.

Since each part has at most one element, there is only one such candidate: $$\{\{1\}, \{2\}, \ldots, \{k\}\}$$

• If $k \leq n,$ then this partition satisfies the requirements of the problem, so there is exactly one such partition.

• If $k > n,$ then this partition has too many parts, so there are zero such partitions.