Question

Seventeen candidates have filed for the upcoming county council election. 7 are women and 10 are men a) Is how many ways can 10 county council members be randomly elected out of the 17 candidates? (b) In how many ways can 10 county council members be randomly elected from 17 candidates if 5 must be women and 5 must be men? c) If 10 county council members are randomly elected from 17 candidates, what is the probability that 5 are women and 5 are men? Round answer to nearest ten-thousandth (4 places after decimal).

Answer 1

Answer: (a) 19448 ways

(b) 5292 ways

(c) 0.2721

Step-by-step explanation:

(a) 10 county council members be randomly elected out of the 17 candidates in the following ways:

= $^{n}C_{r}$

= $^{17}C_{10}$

= $\frac{17!}{10!7!}$

= 19448 ways

(b) 10 county council members be randomly elected from 17 candidates if 5 must be women and 5 must be men in the following ways:

we know that there are 7 women and 10 men in total, so

= $^{7}C_{5}$ × $^{10}C_{5}$

= $\frac{7!}{5!2!}$ × $\frac{10!}{5!5!}$

= 21 × 252

= 5292 ways

(c) Now, the probability that 5 are women and 5 are men are selected:

= $\frac{ ^{7}C_{5} * ^{10}C_{5}}{^{17}C_{10}}$

= $\frac{5292}{19448}$

= 0.2721