Question

A cupcake shop produces 5 types of cupcakes, one of which is chocolate. Assume there is a large amount available of each type of cupcake.

1/ How many different selections of 8 cupcakes are there?

2/ How many different selections of 8 cupcakes have at least 3 chocolate cupcakes?

3/ How many different selections of 8 cupcakes have at most 2 chocolate cupcakes?

Answer 1

Answer: 1) 390625, 2) 1365, 3) 411105.

Explanation:

Since we have given that

Number of types of cupcakes = 5

Number of chocolate type = 1

1/ How many different selections of 8 cupcakes are there?

Number of cupcakes we need to select = 8

Since we have 5 options for each 8 cupcakes.

So, the number of ways to select 8 cupcakes would be :

`5^8=390625`

2/ How many different selections of 8 cupcakes have at least 3 chocolate cupcakes?

Number of different selection would be :

=
`4^5+4^4+4^3+4^2+4+1=1365`

3/ How many different selections of 8 cupcakes have at most 2 chocolate cupcakes?

Number of ways to select no chocolate + Number of ways to select 1 chocolate + Number of ways to select 2 chocolate

`5^8+4^7+4^6=411105`

Hence, 1) 390625, 2) 1365, 3) 411105.