Question

The offices of? president, vice? president, secretary, and treasurer for an environmental club will be filled from a pool of 17 candidates. Nine of the candidates are members of the debate team. ?(a) What is the probability that all of the offices are filled by members of the debate? team? ?(b) What is the probability that none of the offices are filled by members of the debate? team

Answer 1

we are given

total number of candidates =17

Nine of the candidates are members of the debate team

number of debate candidates=9

(a)

Total number of officers=4

we can use combination formula

$nCr=\frac{n!}{r!(n-r)!}$

So, the number of ways in which 4 people can be selected from 17 people is

$17C4=\frac{17!}{4!(17-4)!}$

now, we can simplify it

$17C4=2380$

and the number of ways in which 4 people can be selected from the debate team of 9 is

$9C4=\frac{9!}{4!(9-4)!}$

now, we can simplify it

$9C4=126$

so, P(all offices filled by debate team ​members) is

$=\frac{126}{2380}$

so, P(all offices filled by debate team ​members) is  0.0529..........Answer

(b)

non-debate team is 17-9=8

number of ways in which 4 people can be selected from the non-debate team of 8 is

$8C4=\frac{8!}{4!(8-4)!}$

$8C4=70$

Probability that none of the office is filled by the debate team is

$=\frac{70}{2380}$

$=0.029$..............Answer