Question

An ice cream Parlour has ice creams in eight different varieties. Number of ways of choosing $3$ ice creams taking atleast two ice cremes of the same variety, is ;

• A. $56$
• B. $64$
• C. $100$
• D. $noneofthese$

Answer 1

The correct option is B $64$

There are eight varieties of ice creams inn the ice cream parlour.

We need to choose $3$ creams where atleast two ice creams are of the same variety.

The first possibility is that the three ice creams chosen are of the same variety.

$\therefore$ number of ways of choosing $3$ ice creams which are of the same variety $n_1{=}^8{P}_1=8$

The second possibility is that the two ice creams are of the same variety and the third is of another variety.

$\therefore$ number of ways choosing $2$ ice creams of same variety and third ice cream of another variety, $n_2{=}^8{P}_1{.}^7{P}_1$

$=8.7$

$=56$

Hence, number of ways of choosing $3$ ice creams taking atleast two ice creams of the same variety $=n_1+n_2$

$=8+56$

$=64$