Question

A group consists of fivefive men and fivefive women. ThreeThree people are selected to attend a conference. a. In how many ways can threethree people be selected from this group of tenten​? b. In how many ways can threethree women be selected from the fivefive ​women? c. Find the probability that the selected group will consist of all women.

Answer 1

a).  Three people can be selected from a group of ten in

(10 · 9 · 8) = 720 ways.

(But every group of 3 different people can be selected in 6 different ways.  So there are actually only (720/6) = 120 different groups of 3 different people.)

b).  Three women can be selected from a group of five women in

(5 · 4 · 3) = 60 ways.

(But every group of 3 different women can be selected in 6 different ways.  So there are actually only (60/6) = 10 different groups of 3 different women.)

   

c).  Start with 3 men and 3 women.

The probability of the first selection being a woman is (3/6) = 50% .

Then there are 3 men and 2 women.

The probability of the 2nd selection being a woman is (2/5) = 40% .

Then there are 3 men and 1 woman.

The probability of the 3rd selection being a woman is (1/4) = 25% .

The probability of all 3 selections being women is

(50% ·40% ·25%) = (0.5 ·0.4 ·0.25) = 0.05

The probability that the selected group will consist of all women is 5% .