Chapter 12.4, Problem 24PPS

a.

To determine

To find: the combination that the 12 spots can be chosen.

Answer to Problem 24PPS

The 12 spots can be chosen in 1352078 ways.

Explanation of Solution

Given information:

Total number of students is 23.

Total number of spots is 12.

Formula used:

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

Calculation :

Number of ways to choose the 12 spots are-

  $\begin{array}{l}C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}\\ C\left(23,12\right)=\frac{23!}{\left(23-12\right)!12!}\left[n=23\text{ and }r=12\right]\\ =\frac{23!}{11!12!}\\ =\frac{23\times 22\times 21\times 20\times 19\times 18\times 17\times 16\times 15\times 14\times 13\times 12!}{11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\times 12!}\left[\begin{array}{l}\text{Divide by }\\ \text{common factor}\text{.}\end{array}\right]\\ =1352078\text{ ways}\end{array}$

Hence, there are 1352078 ways that the 12 spots can be chosen.

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 $\frac{1176}{15504}$

b.

To determine

To find: the probability that at least one senior student will be chosen.

Answer to Problem 24PPS

The probability that at least one senior student will be chosen is $\frac{8}{23}$ .

Explanation of Solution

Given information:

Total number of students is $4+5+6+8=23$ .

Total number of senior students is 8.

Calculation :

Total number of non-senior students is $23-8=15$ .

To calculate the probability that at least one senior student will be chosen,

  $\begin{array}{l}\text{P}\left(\text{non-senior students}\right)=\frac{{}^{15}{\text{C}}_1}{{}^{23}{\text{C}}_1}=\frac{15}{23}\\ \text{P}\left(\text{at least one senior student}\right)=1-\frac{15}{23}=\frac{8}{23}\end{array}$

Hence, the probability that at least one senior student will be chosen is $\frac{8}{23}$