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 cards 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` d. `67`

Options

A. 264
B. 265
C. 53
D. 67

Answer

C

Explanation

`because`Card numbered 1 is always placed in envelope numbered 1, we can consider two cases.
Case I Card numbered 2 is placed I envelope numbered 1, then it is derangement of 4 objects, which can be done in
`4!(1-(1)/(1!)+(1)/(2!)-(1)/(3!)+(1)/(4!))=9` ways
Case II card numbered 2 is not placed in envelope numbered 1, then it is derangement of 5 objects, which can be done in
`5!(1-(1)/(1!)+(1)/(2!)-(1)/(3!)+(1)/(4!)-(1)/(5!))=44` ways
`therefore`Total ways=9+44=53 ways.