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 $2$. Then the number of ways it can be done is -

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

Answer 1

The correct option is C $53$

Case (1)
Keep letter2 in box 1 and letter1 in box 2 then apply dearrangements for letter3,4,5,6, we get 9 possibilities
Case(2)
keep letter3 in box 1 and letter1 in box2, now we will be left out with letter2,4,5,6.
now keep letter2 in box3 and take derrangements for letter4,5,6 we get 2 possibilities.
Now keep letter2 in box4 we get 3 possibilities, letter2 in box5 3 possibilities,letter2 in box6 we get 3 possibilities.
Therefore keeping letter3 in box1 and letter1 in box 2 we get 2+3+3+3=11.
Like that similarily for letter4,letter5,letter6
therefore total number ways are 9+11+11+11+11=53