Question

A box contains $2$ white balls and $4$ red balls, in how many ways can three balls be drawn from the box, if at least one red ball is to be included in the draw?

• A. $32$
• B. $48$
• C. $20$
• D. $96$

Answer 1

The correct option is C $20$

We need to pick $3$ balls from $6(2W$ $\xi$ $4R)$ such that we pick at least $1$ red ball.

Since there are only $2$ white balls any combination we pick will have at least one red ball.

Case $1:$ $2W$ $\xi$ $1R$

$\Rightarrow {}^2{C}_2\times {}^4{C}_1=4$

Case $2:$ $1W$ $\xi$ $2R$

$\Rightarrow {}^2{C}_1\times {}^4{C}_2=12$

Case $3:$ All Red

$\Rightarrow {}^4{C}_3=4$

$\Rightarrow$ Total $=4+12+4=20.$

Hence, the answer is $20.$