GATE CS 2023 Set 1 — Question 48
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Engineering Mathematics → Quantitative Aptitude → Permutations & Combinations
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Question
Let , where is a large positive integer greater than 1000. Let be a positive integer less than . Let be subsets of with and . We say that a permutation of separates from if one of the following is true.
- All members of appear in the permutation before any of the members of .
- All members of appear in the permutation before any of the members of .
Correct answer
(D) 2 n2k (n - 2k)! (k!)²
Solution
To find the number of permutations of that separate from :
1.First, choose positions out of available positions in the permutation for the elements of . This can be done in ways.
2.The remaining elements of can be arranged in the remaining positions in ways.
3.Now, we arrange the elements of and elements of in the chosen positions such that they are separated.
- Case 1: All elements of before all elements of . The first positions among the chosen must be filled by elements of ( ways) and the next positions by elements of ( ways). Total: ways.
- Case 2: All elements of before all elements of . Similarly, the first positions are for and the next for . Total: ways.
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