Question

Find the number of ways in which four different letters can be put in their four addressed envelopes so that
$\left(i\right)$ at least two of them are in the wrong envelopes
$\left(ii\right)$ all the letters are in the wrong envelopes

• A. $23,9$
• B. $16,9$
• C. $23,8$
• D. $16,8$

Answer 1

Answer: (A)

$23,9$

There are $4!$ ways of putting 4 letters in 4 envelopes
Atleast two of them in wrong envelope means either,one of them should be wrong placed which will make sure that the other envelope is also wrongly placed(1 way):
$=4!-1$$=23$ ways
None of them are correctly placed$=24-$(correctly placed)
All four are correctly placed = $1$ way
Only three correctly placed = $0$ ways( can not happen as the fourth one will end up in the correct envelope)
Only two correctly placed = ${}^4{C}_2\times 1$ ways(we choose 2 out of 4)
Only one correctly placed =${}^4{C}_1\times 2$ ways(we choose 1 out of 4)
Total=$1+0+6+8$ ways =$15$ ways
So, =$24-15$$=9$ ways
Hence, option 'A' is correct.