Respuesta :
ANSWER
The number of ways of obtaining a committee with exactly 3 Republicans is 7280 ways
STEP-BY-STEP 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
[tex]\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{ }\frac{20!}{12!8!} \\ =\text{ }\frac{20\text{ x 19}\times18\times17\times16\times15\times14\times13\times\cancel{12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1}}{\cancel{12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1!\text{ 8!}}} \\ =\text{ }\frac{20\times19\times18\times17\times16\times15\times14\times13}{8\times7\times6\times5\times4\times3\times2\times1} \\ =\text{ }\frac{5079110400}{40320} \\ =\text{ 125,970 ways} \end{gathered}[/tex]The favorable way of selecting exactly 3 republicans are 14C3(6C3)
[tex]\begin{gathered} ^{14}C_3\text{ = }\frac{14!}{(14-\text{ 3)!3!}} \\ =\text{ }\frac{14!}{11!\text{ 3!}} \\ =\text{ }\frac{14\text{ }\times\text{ 13 }\times12\times}{3\times2\times1} \\ =\text{ }\frac{2184}{6} \\ =\text{ 364 ways} \\ \\ ^6C_3\text{ = }\frac{6!}{(6\text{ - 3)!3!}} \\ =\text{ }\frac{6!}{3!3!} \\ =\text{ }\frac{6\text{ x 5 x 4}}{3\text{ x 2 x 1}} \\ =\text{ }\frac{120}{6} \\ =\text{ 20 ways} \\ 364\text{ x 20 = 7280 ways} \end{gathered}[/tex]Therefore, the number of ways of obtaining a committee with exactly 3 Republicans is 7280 ways
