The planning committee at school has 10 members. Exactly four of these members are teachers. A four-person subcommittee with at least one member who is a teacher must be formed from the members of the planning committee. How many distinct subcommittees are possible?
The first thing I notice is that the subcommittee must have at least one teacher, so I can't choose four non-teachers. That means I have to avoid the combination of choosing four out of the six non-teachers, which is 6 choose 4, or 15. So, the total number of ways to choose four people out of 10, minus the number of ways to choose four non-teachers, should give me the answer. The total number of ways to choose four people out of 10 is 10 choose 4, or 210. Subtracting 15 from 210, I get 195. Therefore, there are 195 distinct subcommittees possible. 