# Combinatorics question about Taking Days Off

A cashier wants to work five days a week, but he wants to have at least one of the Saturday and Sunday off. In how many ways can he choose the days he will work?

So, in this case, what should I count first? How do I start? I know how to solve this if the cashier doesn't want a weekend off, but what do I do if the cashier do?

Thanks!

You can find the answer by subtracting the number of ways that don't work from the total possible number of ways.

First, figure out the total number of ways to choose five days to work out of the seven days of the week, which is ${7\choose5}=21.$

Then, count the ones that would include working on both Saturday and Sunday, which is ${5\choose3}=10.$

$\therefore$ There's $21-10=\boxed{11}$ ways.

• But in that case, what's left is the ways that he would have to work both Saturday and Sunday. I'm subtracting the ones that he will have to work for both Saturday and Sunday. – user67258 Jul 7 '13 at 18:34
• I agree with Danny. +1 – Tomas Jul 7 '13 at 18:36
• I think the wording here's a little confusing, since the title is "...taking days off" but the question is asking for the total possible ways to work 5 days a week. – user67258 Jul 7 '13 at 18:37
• Oops, I was thinking in terms of choosing the days off. Your answer's fine. – Zev Chonoles Jul 7 '13 at 18:38

He either works Monday-Friday, or he chooses one of those days to take off and work Sunday or Saturday instead.

There is one way to work Monday - Friday. There are 5 ways to choose one of the days Monday-Friday, and there are 2 ways to choose a weekend-day instead:

$1 + \binom{1}{5} \cdot 2 = 1 + 5 \cdot 2 = 11$