Question

7 white balls and 3 black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

• A. $\frac{1}{2}$
• B. $\frac{7}{15}$
• C. $\frac{2}{15}$
• D. $\frac{1}{3}$

Answer 1

Answer: (B)

$\frac{7}{15}$

The number of ways of placing $3$ black balls at $10$ places is ${}^{10}{C}_3$.
The number of ways in which two black balls are not together is equal to the number of ways of choosing $3$ places marked with $X$ out of eight places.
$XWXWXWXWXWXWXWX$
This can be done in ${}^8{C}_3$ ways. Thus, probability of the required event is $\frac{{}^8{C}_3}{{}^{10}{C}_3}=\frac{8\times 7\times 6}{10\times 9\times 8}=\frac{7}{15}$