Question

There are four married couples in a club. The number of ways of choosing a committee of three members so that no complete couple appears in the committee is

• A. 4
• B. 8
• C. 16
• D. 32
• E. none of these

Answer 1

The correct option is D 32

Option (d)

The answer ${}^8{C}_3{-}^4{C}_1\times 6$ can be reasoned out as ${}^8{C}_3$ = all the cases wherein out of 8 people any 3 are chosen. Now choosing any three out of eight will include within it cases where even a married couple is chosen. So, we will subtract all those cases wherein married couples are chosen. The total number of arrangements of married couples will be ${}^4{C}_1\times 6$. This is because out of 4 available, we can choose any one married couple. Now along with this married couple the third person chosen will be any of the 6 remaining people. Thus ${}^4{C}_1\times 6$ number of different possibilities need to be subtracted from ${}^8{C}_3$.