An ice cream parlor has 28 different flavors, 8 different kinds of sauce and 12

Question

An ice cream parlor has 28 different flavors, 8 different kinds of sauce and 12 toppings. (a). How many different ways can a dish with 3 scoops of ice cream be made, if each flavor can be used more than once, and the order of the scoops does not matter? (b). How many different kinds of small sundaes are there is a small sundae contains one scoop a sauce and a topping? (c). How many different large sundaes are there if a large sundae contains 3 scoops (flavors can be used more than once, the order of the scoops does not matter) two different kinds of sauce, the order of the sauce does not matter, and three different toppings and the order of the toppings does not matter?

Explanation / Answer

This is a combinaition problem , so we use formula nCr = n!/(n-r)!*r!

a) 3 scoops out of 28 can be done is 28C3 =

C(n,r)=C(28,3)C(n,r)=C(28,3)

=28!(3!(283)!)=28!(3!(283)!)

= 3276

b) one scoop a sauce , one scoop a topping and 28 different flavors can be done is

8C1*12C1 = 8*12*3276(as calculated above) = 314496

c) 28C3 * 8C2 * 12C3

8C2 =

C(n,r)=C(8,2)C(n,r)=C(8,2)

=8!(2!(82)!)=8!(2!(82)!)

= 28

12C3 =

C(n,r)=C(12,3)C(n,r)=C(12,3)

=12!(3!(123)!)=12!(3!(123)!)

= 220

so final answer is

3276*220*28

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