Given:
There are given that the Senate in a certain state is comprised of 59 Republicans, 39 Democrats, and 2 Independents.
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,
[tex](59\text{ choose 3)(39 choose 2)}[/tex]
So,
[tex](59C_3)(39C_2)_{}[/tex]
Then,
From the formula of combination:
[tex]nC_r=\frac{n!}{r!(n-r)!}[/tex]
Then,
[tex](59C_3)(39C_2)_{}=(\frac{59!}{3!(59-3)!})(\frac{39!}{2!(39-2)!}[/tex]
Then,
[tex]\begin{gathered} (59C_3)(39C_2)_{}=(\frac{59!}{3!(59-3)!})(\frac{39!}{2!(39-2)!} \\ =(\frac{59!}{3!(56)!})(\frac{39!}{2!(37)!}) \\ =(\frac{59\times58\times57\times56!}{3!(56)!})(\frac{39\times38\times37!}{2!(37)!}) \\ =(\frac{59\times58\times57}{3!})(\frac{39\times38}{2!}) \end{gathered}[/tex]
Then,
[tex]\begin{gathered} (\frac{59\times58\times57}{3!})(\frac{39\times38}{2!})=(\frac{59\times58\times57}{3\times2\times1})(\frac{39\times38}{2}) \\ =(\frac{59\times58\times57}{6})(\frac{39\times38}{2}) \\ =\frac{289070028}{12} \\ =24089169 \end{gathered}[/tex]
Final answer:
Hence, 24089169 committees can be formed.
