The planning committee at school has 11 members. Exactly five of these members are teachers. A four-person subcommittee with at least one member who is not a teacher must be formed from the members of the planning committee. How many distinct subcommittees are possible?

To determine the number of distinct subcommittees that can be formed, we can use the concept of combinations.

First, let's find the number of ways to form a four-person subcommittee using only teachers. Since there are 5 teachers on the committee, we can choose 4 from them using the combination formula:

nCr = n! / (r!(n-r)!)
where n is the total number of teachers (5) and r is the number of teachers needed for the subcommittee (4).

So, the number of ways to select a four-person subcommittee consisting only of teachers is:
5C4 = 5! / (4!(5-4)!) = 5! / (4! × 1!) = 5

Next, we need to find the number of ways to form a four-person subcommittee with at least one member who is not a teacher. To do this, we can subtract the number of subcommittees consisting only of teachers from the total number of possible subcommittees.

The total number of possible subcommittees can be found using the combination formula again, this time with all 11 members of the planning committee:

11C4 = 11! / (4!(11-4)!) = 11! / (4! × 7!) = 11 × 10 × 9 × 8 / (4 × 3 × 2 × 1) = 330

Finally, subtracting the number of subcommittees consisting only of teachers from the total number of possible subcommittees gives us:
330 - 5 = 325

Therefore, there are 325 distinct subcommittees possible with at least one member who is not a teacher.