A catering service offers 12 appetizers, 9 main courses, and 7 desserts. A customer is to select 6 appetizers, 4 main courses, and 3 desserts for a banquet. In how many ways can this be done?

Answer 1

6 appetizers can be chosen from an available 12 in 924 ways.

This is done by calculating 12!/(6! x 6!) = (12*11*10*9*8*7)/((6*5*4*3*2*1). The symbol ! is called factorial and the idea is that n items can be chosen from a set of s items as s!/(n! * (s-n)!).

Notice that s! can be written as s * s-1 * s-2 * s-3 * ... * s-n+1 * (s-n)!. This allowed us to cancel the (s-n)! from the denominator and shorten the numerator to the first n terms.

4 main courses can be selected from the 9 available in 126 ways using the same concept as above.

3 desserts from 7 available can be made in 7*6*5 / 3*2*1 = 35 ways.

Thus, using the given criteria, appetizers, main courses AND desserts can be chosen in

924 * 126 * 35 = 4,074,840 ways. Wow!