Among the contestants in a competition are 38 women and 22 men. if 5 winners are randomly selected, what is the probability that they are all men

There are two ways to solve this problem:

(1) The basic way:

Total number of contestants = 38 + 22 = 60 contestant

Probability that 1 winner is a man = 22/60

Probability that 2 winners are men = (22/60) * (21/59)

Probability that 3 winners are men = (22/60) * (21/59) * (20/58)

Probability that 4 winners are men = (22/60) * (21/59) * (20/58) * (19/57)

Probability that 5 winners are men = (22/60) * (21/59) * (20/58) * (19/57) * (18/56)

Doing the calculation, we will find that:

probability that 5 winners are men = 0.0048217

(2) The combination way:

Total number of contestants = 38 + 22 = 60

probability that 5 winners are men = C (22, 5) / C (60, 5)

where:

C (n, r) = n! / ((n - r) ! * r!)

(The exclamation mark refers to the factorial)

Now, we will substitute:

C (22,5) = (22) ! / ((22-5) ! * (5) !) = 26334

C (60,5) = (60) ! / ((60-5) ! * (5) !) = 5461512

Substitute to get the probability as follows:

probability that 5 winners are men = C (22, 5) / C (60, 5)

= (26334) / (5461512)

= 0.0048217