Question

The Senate in a certain state is comprised of 59 Republicans, 39 Democrats, and 2 Independents. How many committees can be formed if each committee musthave 3 Republicans and 2 Democrats?

Answer 1

Given:

There are given that the Senate in a certain state is comprised of 59 Republicans, 39 Democrats, and 2 Independents.

Step-by-step explanation:

According to the question, the committees must have 3 Republicans and 2 democrats

That means,

Choose 3 Republicans from 59 Republicans and 2 democrats from 39 Democrats.

The,

`(59\text{ choose 3)(39 choose 2)}`

So,

`(59C_3)(39C_2)_{}`

Then,

From the formula of combination:

`nC_r=(n!)/(r!(n-r)!)`

Then,

`(59C_3)(39C_2)_{}=((59!)/(3!(59-3)!))((39!)/(2!(39-2)!)`

Then,

`\begin{gathered} (59C_3)(39C_2)_{}=((59!)/(3!(59-3)!))((39!)/(2!(39-2)!) \\ =((59!)/(3!(56)!))((39!)/(2!(37)!)) \\ =((59*58*57*56!)/(3!(56)!))((39*38*37!)/(2!(37)!)) \\ =((59*58*57)/(3!))((39*38)/(2!)) \end{gathered}`

Then,

`\begin{gathered} ((59*58*57)/(3!))((39*38)/(2!))=((59*58*57)/(3*2*1))((39*38)/(2)) \\ =((59*58*57)/(6))((39*38)/(2)) \\ =(289070028)/(12) \\ =24089169 \end{gathered}`

Final answer:

Hence, 24089169 committees can be formed.