Question

A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?

$729$

$972$

$1024$

$2187$

$2304$

Answer: A

Concepts:arrangements with restrictions multiplication principle work backwards

Difficulty rating: 1140

Small Hint:

Work backward from Friday

Big Hint:

Each neighboring day has $3$ choices once the next day is fixed

Solution:

We know that there must be cake served on Friday, and as such, on Saturday, we cannot have cake. Therefore, we have $3$ choices for Saturday’s menu.

Furthermore, for each of the $5$ previous days, we could go backwards and have $3$ choices on each day, making the total number of choices $3^5\cdot 3 = 729.$

Thus, the correct answer is A .

2012 AMC 10B Problem 11

Answer: A

Difficulty rating: 1140

Small Hint:

Big Hint:

Solution:

Problem 11 in Other Years