Question

A small production company has 20 employees including 12 men and 8 women. These employees want to form a delegation of 3 members to represent them in an inter-union meeting. How many ways can they form the delegation if it is to contain at least one woman.

Answer 1

A small production company has 20 employees including 12 men and 8 women. These employees want to form a delegation of 3 members to represent them in an inter-union meeting. How many ways can they form the delegation if it is to contain at least one woman

we have that

contain at least one woman.

that means

one woman and two-man

two women and one man

three woman

step 1

Find out the combination 20C3

`20C3=(20!)/(3!(20-3)!)=1,140`

step 2

Find out combination 12C2 and 8C1 (one woman and two-man)

`\begin{gathered} 12C2=(12!)/(2!(12-2)!)=66 \\ 8C1=(8!)/(1!(8-1)!)=8 \end{gathered}`

step 2

Find out the combination two women and one man (8C2 and 12C1)

`\begin{gathered} 8C2=(8!)/(2!(8-2)!)=28 \\ 12C1=(12!)/(1!(12-1)!)=12 \end{gathered}`

step 3

Find out the combination three womwomenC3)

`8C3=(8!)/(3!(8-3)!)=56`

step 4

we have that

the total combination is

66*8+28*12+56=920

the answer is 920