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

Engineering

Mathematics

De arrangements

Question

264

265

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Case-I: 2 goes to 1

$∴$ number of way = D(4)

Case-II: 2 does not goes to 1,

$∴$ number of ways = D(5)

$∴$ answer = D(4) + D(5) = $4! (1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!}) + 5! (1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!})$