At a birthday party a mother prepares a plate of cookies for

At a birthday party, a mother prepares a plate of cookies for 8 children. There are plenty of chocolate chip, peanut butter, and oatmeal cookies, but each child gets only 1 cookie.

a. How many different plates can be prepared?

b. How many different plates can be prepared if at least 1 of each kind of cookie is given out?

c. How many different plates can be prepared if no one likes oatmeal cookies?

d. How many different plates can be prepared if 2 children insist on getting peanut butter?

e. How many different plates can be prepared if the dog got into the kitchen and ate all the chocolate chip cookies except 2?

Solution

a) The plate contains 8 cookies, in the first case there is no restriction on the cookies choosen, hence the first cookie can be either of chocolate chip, peanut butter or oatmal cookies, similar so on, so we have 3 possibilities for every cookies

Hence the number of cookies will be equal to

=> 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3

=> 6561 ways

c) Different plates can be prepared if no one likes oatmeal cookies

Hence there will be only 2 possible combinations for all the 8 cookies given to the students

Hene number of plates possible

=> 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2

=> 256 ways

d) Number of different plates since 2 childrens only wanted peanut cookies, hence it can be done in only one way for 2 students and remaining 6 students can take any of the three possible cookies

=> 3 * 3 * 3 * 3 * 3 * 3

=> 729 ways

e) The number of chocolate chip cookies remaining is equal to 2

Hence the first two cookies can be either chocolate or peanut butter or oatmeal cookies

=> 3 * 3 * 2 * 2 * 2 * 2 * 2 * 2

=> 9 * 64

=> 576 ways