Question:
If $ ^{32}{{P}_{6}}=k\,{{(}^{32}}{{C}_{6}}), $ then $k$ is equal to
If $ ^{32}{{P}_{6}}=k\,{{(}^{32}}{{C}_{6}}), $ then $k$ is equal to
Updated On: Jun 23, 2024
- $ 6 $
- $ 24 $
- $ 120 $
- $ 720 $
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The Correct Option is D
Solution and Explanation
Given, $ ^{32}{{P}_{e}}=k\,{{(}^{32}}{{C}_{6}}) $
$ \Rightarrow $ $ \frac{32!}{(32-6)!}=k.\frac{32!}{6!(32-6)!} $ $ \left[ \because \,{{\,}^{n}}{{P}_{r}}=\frac{n!}{(n-r)!}\,\,and{{\,}^{n}}{{C}_{r}}=\frac{n!}{r!(n-r)!} \right] $
$ \Rightarrow $ $ 1=\frac{k}{6!}\Rightarrow k=6! $
$ \Rightarrow $ $ k=6\times 5\times 4\times 3\times 2\times 1=720 $
$ \Rightarrow $ $ \frac{32!}{(32-6)!}=k.\frac{32!}{6!(32-6)!} $ $ \left[ \because \,{{\,}^{n}}{{P}_{r}}=\frac{n!}{(n-r)!}\,\,and{{\,}^{n}}{{C}_{r}}=\frac{n!}{r!(n-r)!} \right] $
$ \Rightarrow $ $ 1=\frac{k}{6!}\Rightarrow k=6! $
$ \Rightarrow $ $ k=6\times 5\times 4\times 3\times 2\times 1=720 $
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Concepts Used:
Permutations and Combinations
Permutation:
Permutation is the method or the act of arranging members of a set into an order or a sequence.
- In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point.
- A permutation is used in many events of daily life. It is used for a list of data where the data order matters.
Combination:
Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.
- Combination refers to the combination of about n things taken k at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.