Question

question 1
3 pts
a catering service offers 8 appetizers, 7 main courses, and 7 desserts. a company wants to hire the catering service for a banquet it intends to hold for its employees. the company would like to select 3 appetizers, 4 main courses, and 3 desserts for the banquet. in how many different ways can this be done? (enter your answer in the box below, without using comma in the number. the program will insert comma in the number where needed when you submit your answer).

Explanation:

Step1: Calculate the number of ways to choose appetizers

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n\) is the total number of items and \(r\) is the number of items to be chosen.
For appetizers, \(n = 8\) and \(r=3\).

$$C(8,3)=\frac{8!}{3!(8 - 3)!}=\frac{8!}{3!5!}=\frac{8\times7\times6}{3\times2\times1}=56$$

Step2: Calculate the number of ways to choose main courses

For main courses, \(n = 7\) and \(r = 4\)

$$C(7,4)=\frac{7!}{4!(7 - 4)!}=\frac{7!}{4!3!}=\frac{7\times6\times5}{3\times2\times1}=35$$

Step3: Calculate the number of ways to choose desserts

For desserts, \(n=7\) and \(r = 3\)

$$C(7,3)=\frac{7!}{3!(7 - 3)!}=\frac{7!}{3!4!}=\frac{7\times6\times5}{3\times2\times1}=35$$

Step4: Use the multiplication principle

The total number of ways to make the selection is the product of the number of ways of choosing each type of food.

$$N = C(8,3)\times C(7,4)\times C(7,3)$$

$$N=56\times35\times35$$

$$N = 56\times1225$$

$$N=68600$$

Answer:

68600