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 ___________.
Solution:
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
$=\frac{{ }^{3} C_{3}}{\text { all black }}+\frac{{ }^{3} C_{2} \times{ }^{6} C_{1}}{2 \text { black balls }}+\frac{{ }^{3} C_{1} \times{ }^{6} C_{2}}{2 \text { black balls }}$
$=1+3 \times 6+3 \times \frac{6 \times 5}{5}$
$=1+18+45$
$=64$
i.e The number of ways three balls be drawn from the box with at-lest one black ball included is 64.