Permutations and Combinations
Derangements
JEE Advanced 2014 Paper 2
Grade 11
Question:
Six cards and six envelopes are numbered $1, 2, 3, 4, 5, 6$. Cards are placed in envelopes so that each envelope has exactly one card and no card is in the same-numbered envelope. Card 1 is always placed in envelope 2. The number of such arrangements is:
(1) $264$
(2) $265$
(3) $53$
(4) $67$
Step-by-Step Solution
Key Concept: Fix card 1 in envelope 2. Remaining cards $\{2,3,4,5,6\}$ go to envelopes $\{1,3,4,5,6\}$. Card 2's forbidden envelope (2) is already used, so only cards 3,4,5,6 have active restrictions. By inclusion-exclusion: $5! - \binom{4}{1}4! + \binom{4}{2}3! - \binom{4}{3}2! + \binom{4}{4}1! = 120 - 96 + 36 - 8 + 1 = 53$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)