Question

A closet has $5$ pairs of shoes. The number of ways in which $4$ shoes can be drawn from it such that there will be no complete pair is?

• A. $80$
• B. $160$
• C. $200$
• D. $Noneoftheforegoingnumbers$

Answer 1

The correct option is A $80$

Total shoes $\Rightarrow$ 5 pair of shoes

= Total ways of selecting 4 pair out of 5

$\Rightarrow {(}^5{C}_4)$

Now for every pair, on selecting 1 shoe,

this can be done in ${(}^2{C}_1{)}^4$ ways

$\therefore$ Total no. of ways = ${(}^5{C}_4){(}^2{C}_1{)}^4$

$\Rightarrow \left(\frac{5\times 4!}{4!}\right)\times (\frac{2\times 1}{1!}{)}^4$

$=5\times 2^4$

$=5\times 16=80$ ways (Option A)