Question

The number of ways in which $5$ balls can be selected from a bag containing $5$ identical and $5$ different balls is

• A. $24$
• B. $30$
• C. $32$
• D. $36$

Answer 1

The correct option is C $32$

Let the balls be $a,a,a,a,a,b_1,b_2,b_3,b_4,b_5$. The number of ways of selecting $r$ identical balls is one. The number of ways of selecting 5-r balls from the 5 different balls is $\left(\begin{array}{cc}5& \\ 5-& r\end{array}\right)=\left(\begin{array}{c}5\\ r\end{array}\right)$ The desired number is $\sum _{r=0}^5\left(\begin{array}{c}5\\ r\end{array}\right)=2^5=32$