GATE CS 2017 Set 1 — Question 64
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General Aptitude → Quantitative Aptitude → Permutations & Combinations
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Question
Arun, Gulab, Neel and Shweta must choose one shirt each from a pile of four shirts coloured red, pink, blue and white respectively. Arun dislikes the colour red and Shweta dislikes the colour white. Gulab and Neel like all the colours. In how many different ways can they choose the shirts so that no one has a shirt with a colour he or she dislikes?
Correct answer
(D) 14
Solution
Total number of ways to distribute 4 distinct shirts to 4 people is .Let the people be A, G, N, S and shirts be R, P, B, W.
Constraints:
Let be the set of permutations where A chooses R.
Let be the set of permutations where S chooses W.We want to find the number of valid permutations, which is .Using the Inclusion-Exclusion Principle:
.
.
.Now, .The number of valid ways is .
Constraints:
1.A cannot choose R.
2.S cannot choose W.
Let be the set of all permutations. .Let be the set of permutations where A chooses R.
Let be the set of permutations where S chooses W.We want to find the number of valid permutations, which is .Using the Inclusion-Exclusion Principle:
1.Calculate (A gets R):
A is fixed to R. The remaining 3 people (G, N, S) can be arranged in ways..
2.Calculate (S gets W):
S is fixed to W. The remaining 3 people (A, G, N) can be arranged in ways..
3.Calculate (A gets R AND S gets W):
A is fixed to R, and S is fixed to W. The remaining 2 people (G, N) can be arranged in ways..Now, .The number of valid ways is .
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