# Choices for course selection

I read this question on brilliant.org:

Winston must choose 4 courses for his final semester of school. He must take at least 1 science class and at least 1 arts class. If his school offers 4 science classes, 3 arts classes and 3 other classes, how many different choices for classes does he have?

My solution: His schedule can be written as {Science, Arts, Any, Any}, in which:

• there are $4$ choices for the first Science class
• there are $3$ choices for the first Arts class
• Now that the requirements are satisfied, we can pool the rest of the classes, giving us $2$ slots for $3+(4-1) +(3-1) = 8$ classes, which can be filled in $8\times 7$ ways.

In total, he has $4\times 3\times 8\times 7 = 672$ ways to choose his classes. However, the website marked my answer as incorrect.

What's the correct answer and why?

• Probably {Physics, Painting, Biology, Chemistry} is not a different schedule from {Biology, Painting, Chemistry, Physics}, but your enumeration scheme counts them as distinct. – Austin Mohr Jul 15 '13 at 4:14

If there is to be no science class, then we can choose our four classes from the remaining six options in $\binom{6}{4}$ ways.
If there is to be no arts class, then we can choose our four classes from the remaining seven options in $\binom{7}{4}$ ways.
Now, there are $\binom{6}{4} + \binom{7}{4}$ schedules that lack either a science class or an arts class. These are the bad schedules. We want to subtract this from the total number of unrestricted schedules, of which there are $\binom{10}{4}$.
Finally, the number of schedules that have at least one science class and at least one arts class is $$\binom{10}{4} - \binom{7}{4} - \binom{6}{4} = 160.$$