Question

A committee of eight Representatives will be selected from a group of fourteenRepublicans and six Democrats. Find the number of ways of obtaining a committee withexactly three Republicans.

Answer 1

The number of ways of obtaining a committee with exactly 3 Republicans is 7280 ways

Explanation:

Given information

A committee of 8 representatives will be selected from a group of 14 Republicans and six Democrats.

From the above information

The total number of persons = 14 + 6

The total number of persons = 20

The total number of ways of selecting 8 representatives is 20C8

`\begin{gathered} ^(20)_{}C_8 \\ \text{Recall that, } \\ ^nC_r\text{ = }\frac{n!}{(n\text{ - r)!r!}} \\ ^(20)C_8\text{ = }\frac{20!}{(20\text{ -8)!8!}} \\ =\text{ }(20!)/(12!8!) \\ =\text{ }\frac{20\text{ x 19}*18*17*16*15*14*13*\cancel{12*11*10*9*8*7*6*5*4*3*2*1}}{\cancel{12*11*10*9*8*7*6*5*4*3*2*1!\text{ 8!}}} \\ =\text{ }(20*19*18*17*16*15*14*13)/(8*7*6*5*4*3*2*1) \\ =\text{ }(5079110400)/(40320) \\ =\text{ 125,970 ways} \end{gathered}`

The favorable way of selecting exactly 3 republicans are 14C3(6C3)

`\begin{gathered} ^(14)C_3\text{ = }\frac{14!}{(14-\text{ 3)!3!}} \\ =\text{ }\frac{14!}{11!\text{ 3!}} \\ =\text{ }\frac{14\text{ }*\text{ 13 }*12*}{3*2*1} \\ =\text{ }(2184)/(6) \\ =\text{ 364 ways} \\ \\ ^6C_3\text{ = }\frac{6!}{(6\text{ - 3)!3!}} \\ =\text{ }(6!)/(3!3!) \\ =\text{ }\frac{6\text{ x 5 x 4}}{3\text{ x 2 x 1}} \\ =\text{ }(120)/(6) \\ =\text{ 20 ways} \\ 364\text{ x 20 = 7280 ways} \end{gathered}`

Therefore, the number of ways of obtaining a committee with exactly 3 Republicans is 7280 ways