Question

catering service offers 8 ​appetizers, 11 main​ courses, and 7 desserts. A banquet committee is to select 7 ​appetizers, 8 main​ courses, and 4 desserts. How many ways can this be​ done?

Answer 1

Answer:  The required number of ways is 46200.

Step-by-step explanation:  Given that a catering service offers 8 ​appetizers, 11 main​ courses, and 7 desserts.

A banquet committee is to select 7 ​appetizers, 8 main​ courses, and 4 desserts.

We are to find the number of ways in which this can be done.

We know that

From n different things, we can choose r things at a time in $^nC_r$ ways.

So,

the number of ways in which 7 appetizers can be chosen from 8 appetizers is

$n_1=^8C_7=\dfrac{8!}{7!(8-7)!}=\dfrac{8\times7!}{7!\times1}=8,$

the number of ways in which 8 main courses can be chosen from 11 main courses is

$n_2=^{11}C_8=\dfrac{11!}{8!(11-8)!}=\dfrac{11\times10\times9\times8!}{8!\times3\times2\times1}=165$

and the number of ways in which 4 desserts can be chosen from 7 desserts is

$n_3=^7C_4=\dfrac{7!}{4!(7-4)!}=\dfrac{7\times6\times5\times4!}{4!\times3\times2\times1}=35.$

Therefore, the number of ways in which the banquet committee is to select 7 ​appetizers, 8 main​ courses, and 4 desserts is given by

$n=n_1\times n_2\times n_3=8\times165\times35=46200.$

Thus, the required number of ways is 46200.