a four - person committee is chosen from a group of eight boys and six girls. if students are chosen at…

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?\n\\(\\frac{4}{1001}\\)\n\\(\\frac{15}{1001}\\)\n\\(\\frac{10}{143}\\)\n\\(\\frac{133}{143}\\)

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?\n\\(\\frac{4}{1001}\\)\n\\(\\frac{15}{1001}\\)\n\\(\\frac{10}{143}\\)\n\\(\\frac{133}{143}\\)

Answer

Explanation:

Step1: Calculate total number of ways to choose 4 - person committee

The total number of students is (8 + 6=14). The number of ways to choose a 4 - person committee from 14 students is given by the combination formula (C(n,r)=\frac{n!}{r!(n - r)!}), where (n = 14) and (r = 4). [C(14,4)=\frac{14!}{4!(14 - 4)!}=\frac{14!}{4!×10!}=\frac{14\times13\times12\times11}{4\times3\times2\times1}=1001]

Step2: Calculate number of ways to choose 4 - boy committee

The number of ways to choose 4 boys from 8 boys is given by the combination formula with (n = 8) and (r = 4). [C(8,4)=\frac{8!}{4!(8 - 4)!}=\frac{8!}{4!×4!}=\frac{8\times7\times6\times5}{4\times3\times2\times1}=70]

Step3: Calculate the probability

The probability (P) that the committee consists of all boys is the number of favorable outcomes (choosing 4 boys) divided by the number of total outcomes (choosing any 4 - person committee). [P=\frac{C(8,4)}{C(14,4)}=\frac{70}{1001}=\frac{10}{143}]

Answer:

(\frac{10}{143})