Question

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, the number of ways it can be done is -

• A. 264
• B. 265
• C. 53
• D. 67

Answer 1

The correct option is C 53

Since, card numbered 1 is always placed in envelope numbered 2, we can consider two cases.
Case I Card numberd 2 is placed in envelope numbered 1. Then it is dearrangement of 4 objects, which can be done in $4!(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!})=9$
Case II Card numbered 2 is not placed in envelope numbered 1.
Then it is dearrangement of 5 objects, which can be done in $5!(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!})=44$ ways.
$\therefore$ Total ways = 44 + 9 = 53