Question

In how many ways $5$ letters go into $5$ envelopes such that exactly $1$ letter goes in to corresponding envelope?

• A. $141$
• B. $142$
• C. $143$
• D. $145$

Answer 1

Answer: (B)

$142$

$5$ letters can go into $5$ envelopes in $5!$ ways, i.e. $5!$ = $5\times 4\times 3\times 2\times 1\times$ = $20\times 6$= $120$
and there is just $1$ combination in which each and every letter goes to its corresponding envelope, thus $120-1=119$ for first condition
$1$ letter in its correct corresponding envelope and $4$ others juggled, here we get $4!$ = = $4\times 3\times 2\times 1\times$ = $24$ again we should subtract 1 combination in which all 4 letters can go correctly to their respective envelopes, thus $24-1$ = $23$ for Second condition.
= $119+23=142$ ways.