probability question

Question

A group consists of six men and seven women. Six people are selected to attend a conference.
a. In how many ways can six people be selected from this group of thirteen ?
b. In how many ways can six women be selected from the seven  women?
c. Find the probability that the selected group will consist of all women.

Answer 1

This is a combinations problem since the order the people are chosen doesn't matter as long as you have a group of six.

 

In each setup, you have a total number of people to choose from (n) and are choose in certain number of people to be a subgroup (r).

 

The number of ways to pick r items from a set of n is _nC_r = n!/[(n-r)!r!].  (Note some calculators allow you to calculate _nC_r directly and most can calculate factorials ! directly.)

 

In part a, you want to pick 6 people from the 13, so n = 13 and r = 6.  ₁₃C₆ = 13!/[(13-6)!6!] = 13!/(7!6!) = 1716.

 

In part b, you want to pick 6 women from the 7 possible, so n = 7 and r = 6.  ₇C₆ = 7!/[(7-6)!6!] = 7!/(1!6!) = 7

 

In part c, we note that of all 1716 possible groups of 6, only 7 of them will contain only women.  Therefore the probability of only selecting all women it 7/1716 ≈ 0.004.

 

(Note: if you looked at the probability of picking a group of all men, it would only be 1/1716 since there is only one group of all 6 men that can be made.  It is far more likely that a mixed gender group will be picked.)