Question

A box contain 2 white balls, 3 black balls and 4 red balls. The number of ways three balls be drawn from the box if at least one black ball is to be included in the draw is ___________.

Answer 1

At-least 1 black ball can be selected in following ways
1 black ball and two non - black or 2 black ball and one non - black or all 3 black balls
∴ Total number of ways of selecting is
$$\begin{gathered} =\frac{{}^{3}C_3}{\mathrm{all}\mathrm{black}}+\frac{{}^{3}C_2\times {}^{6}C_1}{2\mathrm{black}\mathrm{balls}}+\frac{{}^{3}C_1\times {}^{6}C_2}{2\mathrm{black}\mathrm{balls}} \\ =1+3\times 6+3\times \frac{6\times 5}{5} \\ =1+18+45 \\ =64 \end{gathered}$$

i.e The number of ways three balls be drawn from the box with at-lest one black ball included is 64.