Question

a four - person committee is chosen from a group of eight boys and six girls. if students are chosen at random, what is the probability that the committee consists of all boys
\\(\frac{4}{1001}\\)
\\(\frac{15}{1001}\\)
\\(\frac{10}{143}\\)
\\(\frac{133}{143}\\)

Explanation:

Step1: Calculate total student count

Total students = 8 boys + 6 girls = 14

Step2: Compute total 4-person committees

Use combination formula $C(n,k)=\frac{n!}{k!(n-k)!}$.
Total committees: $C(14,4)=\frac{14!}{4!(14-4)!}=\frac{14\times13\times12\times11}{4\times3\times2\times1}=1001$

Step3: Compute all-boy 4-person committees

All-boy committees: $C(8,4)=\frac{8!}{4!(8-4)!}=\frac{8\times7\times6\times5}{4\times3\times2\times1}=70$

Step4: Calculate probability

Probability = $\frac{\text{All-boy committees}}{\text{Total committees}}=\frac{70}{1001}=\frac{10}{143}$

Answer:

$\boldsymbol{\frac{10}{143}}$ (Option: $\boldsymbol{\frac{10}{143}}$)