JEE Advanced · 2014 · 25 May 2014 · Paper 2 · Permutations and Combinations

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Six cards and six envelopes are numbered 1,2,3,4,5,6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is

(A)264
(B)265
(C)53✓ Correct
(D)67

Answer: (C)

Solution

The total number of derangements of 6 objects is:

D_{6} = 6! (1 - \frac{1}{1 !} + \frac{1}{2 !} - \frac{1}{3 !} + \frac{1}{4 !} - \frac{1}{5 !} + \frac{1}{6 !}) = 265

Since card 1 is fixed in envelope 2, and by symmetry card 1 can go into any of the 5 wrong envelopes with equal probability, the number of derangements where card 1 goes specifically into envelope 2 is:

\frac{265}{5} = 53

JEE Advanced · 2014 · 25 May 2014 · Paper 2 · Single correct option

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JEE Advanced · 2014 · 25 May 2014 · Paper 2 · Permutations and Combinations

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