Practice Problems Week 5

You may collaborate and are encouraged to collaborate with your peers. If you do, be sure to understand the material and write it out in your own words.

There is no requirement to submit the material, but this will help you solidify the concepts.

Instructions:

The door to a building has a lock which has 5 buttons numbered from 1 to 5. The combinations of numbers that opens the lock is a sequence of 5 numbers and is reset every week.

a. How many combinations are possible if every button must be used once?

b. Assume that the lock can also have combinations that require you to push 2 buttons simultaneously, and then the other three one at a time. How many combinations does this permit? To clarify, in a sequence of 5 buttons, 2 consecutive buttons must be pressed at the same time to be considered correct.
A computer has 3 processors that receive n tasks. Tasks are assigned to the processors purely at random, meaning that all $3^n$ possible assignments are equally likely. How many possible assignments are there where exactly one processor has no task assigned?
In how many ways can we choose five people from a group of ten to form a basketball team?
Show that $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$
Someone wants to color their fingernails on one hand using at most 2 of the colors red, yellow, and blue. How many ways can they do this?