A student has some $1 and $5 bills in his wallet. He has a total of 31 bills that are worth $33. How many of each type of bill does he have?

There can't be more than seven $5 bills, because they would be worth $33.

So, let's try all other cases and see which fits the requests:

If there are six $5 bills, they are worth $30. You need three more $1 bills to reach $33. So, you have a total of 9 bills, which is not what we want. If there are five $5 bills, they are worth $25. You need eight more $1 bills to reach $33. So, you have a total of 13 bills, which is not what we want. If there are four $5 bills, they are worth $20. You need thirteen more $1 bills to reach $33. So, you have a total of 17 bills, which is not what we want. If there are three $5 bills, they are worth $15. You need eighteen more $1 bills to reach $33. So, you have a total of 21 bills, which is not what we want. If there are two $5 bills, they are worth $10. You need twenty-three more $1 bills to reach $33. So, you have a total of 25 bills, which is not what we want. If there is one $5, it is worth $5. You need twenty-eight more $1 bills to reach $33. So, you have a total of 29 bills, which is not what we want.

So, there's no way you can have 31 bills worth $1 and $5 that are worth $33 in total.