Chapter 12.4, Problem 23PPS

To determine

To find: the probability that 5 marbles of each color are selected.

Answer to Problem 23PPS

The probability that 5 marbles of each color are selected is $\frac{1176}{15504}$ .

Explanation of Solution

Given information:

Total number of marbles is 20.

Total number of selected marbles is 15.

Total number of red marbles is 7.

Total number of purple marbles is 8.

Total number of green marbles is 5.

Formula used:

  $C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}$

Calculation :

To calculate the probability that 5 marbles of each color are selected, there are ${}^{20}{C}_{15}$ ways to choose 15 marbles randomly from the 20 marbles

  $\begin{array}{l}C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}\\ C\left(20,15\right)=\frac{20!}{\left(20-15\right)!15!}\left[n=20\text{ and }r=15\right]\\ =\frac{20!}{5!15!}\\ =\frac{20\times 19\times 18\times 17\times 16\times 15!}{5\times 4\times 3\times 2\times 1\times 5!}\left[\text{Divide by common factor}\text{.}\right]\\ =15504\text{ ways}\end{array}$

Considering that there are 7 red marbles, there are ${}^7{C}_5$ ways to choose the red marbles.

Likewise, there are ${}^8{C}_5$ ways to choose purple marbles and ${}^5{C}_5$ ways to choose green marbles.

Thus, the probability that 5 marbles of each color are selected is-

  $\begin{array}{l}\frac{{}^7{C}_5{\times }^8{C}_5{\times }^5{C}_5}{{}^{20}{C}_{15}}\\ =\frac{21\times 56\times 1}{15504}\\ =\frac{1176}{15504}\end{array}$

Hence, the probability that 5 marbles of each color are selected is $\frac{1176}{15504}$