Question:
If $ ^{m+n}P{{ }_{2}}=90 $ and $ ^{m-n}P{{ }_{2}}=30, $ then $ (m,n) $ is given by (where, m and n are positive integers)
If $ ^{m+n}P{{ }_{2}}=90 $ and $ ^{m-n}P{{ }_{2}}=30, $ then $ (m,n) $ is given by (where, m and n are positive integers)
Updated On: Jun 23, 2024
- $ (8,\,2) $
- $ (5,\,\,6) $
- $ (3,\,\,7) $
- $ (8,\,\,3) $
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The Correct Option is A
Solution and Explanation
By taking option [a], $ (m,\,n)=(8,\,2) $
Now, $ ^{m+n}{{P}_{2}}{{=}^{8+2}}{{P}_{2}}{{=}^{10}}{{P}_{2}} $
$ =\frac{10\,!}{8!}=90 $
and $ ^{m-n}{{P}_{2}}{{=}^{8-2}}{{P}_{2}}{{=}^{6}}{{P}_{2}} $
$ =\frac{6!}{4!}=30 $
Now, $ ^{m+n}{{P}_{2}}{{=}^{8+2}}{{P}_{2}}{{=}^{10}}{{P}_{2}} $
$ =\frac{10\,!}{8!}=90 $
and $ ^{m-n}{{P}_{2}}{{=}^{8-2}}{{P}_{2}}{{=}^{6}}{{P}_{2}} $
$ =\frac{6!}{4!}=30 $
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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.