First slide

Combinations

Question

A box contains two white balls, three black balls and four red balls. The number of ways in which three balls can be drawn from the box if atleast one black ball is to be included in the draw, is

Moderate

A

32
B

64
C

128
D

None of these

Solution

The number of ways of selecting 3 balls out of total 9 (2 white, 3 black, 4 red balls) is $^{9} C_{3}$
i.e., $\frac{9 \times 8 \times 7}{6} = 84$
The number of ways of selecting 3 balls out of non-black six balls is $^{6} C_{3}$
i.e., $\frac{6 \times 8 \times 4}{3 \times 2 \times 1} = 20$
Therefore, the number of ways of selecting 3 balls out of 9 balls so as to include atleast one black ball = 84 – 20 = 64.

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