Respuesta :
Answer: 3,465
Step-by-step explanation:
Given, Number of Democrats= 11
Number of Republicans = 7
Total persons in the group =18
Number of congressman will be selected from the group = 8
Now , according to the combinations , the number of ways of obtaining a committee with exactly 3 Democrats.= (3 Democrats , 5 Republicans)
[tex]=^{11}C_3\times^7C_5\\\\=\dfrac{11!}{3!8!}\times\dfrac{7!}{5!2!}\\\\=\dfrac{11\times10\times9\times8!}{6\times8!}\times\dfrac{7\times6\times5!}{5!\times2}\\\\=165\times21=3465[/tex]
Hence, the number of ways of obtaining a committee with exactly 3 Democrats. = 3,465
Answer:
[tex]3465[/tex]
Step-by-step explanation:
GIVEN: A committee of eight congressman will be selected from a group of [tex]11[/tex] Democrats and [tex]7[/tex] Republicans.
TO FIND: The number of ways of obtaining a committee with exactly [tex]3[/tex]Democrats.
SOLUTION:
Total number of congressman in committee [tex]=8[/tex]
Total number of republicans available [tex]=7[/tex]
Total number of democrats available [tex]=11[/tex]
As, the committee should have exactly [tex]3[/tex] democrats, there will be [tex]5[/tex] republicans in committee
No. of ways of selecting [tex]3[/tex] democrats out of [tex]11[/tex] democrats [tex]^1^1C_3=\frac{11!}{(11-3)!3!}[/tex]
[tex]=\frac{11!}{8!3!}[/tex]
[tex]=165[/tex]
NO. of ways of selecting [tex]5[/tex] republicans out of [tex]7[/tex] republicans [tex]^7C_5=\frac{7!}{(7-5)!5!}[/tex]
[tex]=\frac{7!}{5!2!}[/tex]
[tex]=21[/tex]
The number of ways of obtaining a committee with exactly [tex]3[/tex] Democrats.
[tex]=\text{total ways of selecting 3 democrats}\times\text{total ways of selecting 5 republicans}[/tex]
[tex]=165\times21[/tex]
[tex]=3465[/tex]
Hence there are [tex]3465[/tex] ways to form a committee in which there are exactly [tex]3[/tex] democrats.
