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  1. The six faces of a cube are painted black. The cube is then cut into 53 or 125 smaller cubes, all the same size. How many of the smaller cubes do not have any black faces?
  2. I run a book club with n people, not including myself. Every day, for 365 days, I invite three members in the club to review a book. What is the smallest positive integer n so that I can avoid ever having the exact same group of three members over all 365 days?
  3. How many ways can you distribute 4 identical balls among 4 identical boxes?

  4. There are twice as many students in the cooking club as in the drama club. Suppose there are a students in the drama club and b students who are members of both clubs. Find an expression for the total number of students who are in the cooking club or the drama club (or both). Give your answer in simplest form.

 Nov 10, 2022
 #1
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1. The answer is 64.

 

2. The smallest n that works is 19.

 Nov 10, 2022
 #2
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3. There are 23 ways.

 

4. The answer is 5a - 2b.

 Nov 10, 2022

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