Question

A bag contains 10 different balls. Five balls are drawn simultaneously and then replaced and then seven balls are drawn. The probability that exactly three balls are common to the two drawn is

• A. $\frac{5}{12}$
• B. $\frac{5}{24}$
• C. $\frac{3}{5}$
• D. $\frac{7}{12}$

Answer 1

Answer: (A)

The number of ways of drawing 7 balls (second draws) $={ }^{10} C _7$.
For each set of 7 balls of the second draw 3 must be common to the set of 5 balls of the first draw, i.e., 2 other balls can be drawn in ${ }^3 C _2$ ways.
Thus, for each set of 7 balls of the second draw, there are ${ }^7 C _3 \times{ }^3 C _2$ ways of making the first draw so that there are 3 balls common.
Hence, the probability of having 3 balls in common
$=\frac{{ }^7 C _3 \times{ }^3 C _2}{{ }^{10} C _7}=\frac{5}{12}$