Question

A company must select 4 candidates to interview from a list of 12, which consist of 8 men and 4 women.

How many selections of 4 candidates regardless of gender?
How many selections are possible, if 2 women must be selected?
How many selections are possible, if at least 2 women must be selected?

Answer 1

Selections of 4 candidates regardless of gender would result in 12c4, or 12!/(8!*4!) = 9*10*11*12 = 495 possibilities. Selections of 4 candidates where exactly two must be women would result in 8c2 (for the men) * 4c2 (for the women) = 8!/(6!*2!) * 4!/(2!*2!) = 28 * 6 = 168 possibilities. Selections of 4 candidates where AT LEAST two must be women would result in the above 168 (2 women) plus groups where there are exactly three women (8c1*4c3 = 8*(4!/(3!*1!) = 8*4 = 32) plus groups where there are exactly 4 women (1). So there are 168 + 32 + 1 = 201 possible selections of 4 from this group where at least 2 are women.

Answer 2

Part A

If 4 candidates were to be selected regardless of gender, that means that 4 candidates is to be selected from 12.

The number of possible selections of 4 candidates from 12 is given by


`^(12)C_4= (12!)/(4!(12-4)!)= (12!)/(4!*8!) =11*5*9=495`

Therefore, the number of selections of 4 candidates regardless of gender is 495.



Part B:


If 4 candidates were to be selected such that 2 women must be selected, that means that 2 men candidates is to be selected from 8 and 2 women candidates is to be selected from 4.

The number of possible selections of 2 men candidates from 8 and 2 women candidates from 4 is given by


`^(8)C_2* ^(4)C_2= (8!)/(2!(8-2)!)* (4!)/(2!(4-2)!) \\ \\ = (8!)/(2!*6!)*(4!)/(2!*2!) =4*7*2*3=168`

Therefore, the number of selections of 4 candidates such that 2 women must be selected is 168.



Part 3:

If 4 candidates were to be selected such that at least 2 women must be selected, that means that 2 men candidates is to be selected from 8 and 2 women candidates is to be selected from 4 or 1 man candidates is to be selected from 8 and 3 women candidates is to be selected from 4 of no man candidates is to be selected from 8 and 4 women candidates is to be selected from 4.

The number of possible selections of 2 men candidates from 8 and 2 women candidates from 4 of 1 man candidates from 8 and 3 women candidates from 4 of no man candidates from 8 and 4 women candidates from 4 is given by


`^(8)C_2* ^(4)C_2+ ^(8)C_1* ^(4)C_3+ ^(8)C_0* ^(4)C_4 \\ \\ = (8!)/(2!(8-2)!)* (4!)/(2!(4-2)!)+(8!)/(1!(8-1)!)* (4!)/(3!(4-3)!)+(8!)/(0!(8-0)!)* (4!)/(4!(4-4)!) \\ \\ = (8!)/(2!*6!)*(4!)/(2!*2!)+(8!)/(1!*7!)*(4!)/(3!*1!)+(8!)/(0!*8!)*(4!)/(4!*0!) \\ \\ =4*7*2*3+8*4+1*1=168+32+1=201`

Therefore, the number of selections of 4 candidates such that at least 2 women must be selected is 201.