For a fishing trip, Henry is going to choose lures to put in his tackle box. He has 9 lures that are spinnerbaits and 10 that are buzzbaits. In how many ways can he choose 8 lures if fewer than 3 must be buzzbaits?

To solve this problem, we need to consider two scenarios: when Henry chooses 0, 1, or 2 buzzbaits.

1. If Henry chooses 0 buzzbaits, he must choose 8 spinnerbaits. There is only 1 way to do this, because he has 9 spinnerbaits and he needs to choose 8.

2. If Henry chooses 1 buzzbait, he must choose 7 spinnerbaits. The number of ways to do this is the number of ways to choose 1 out of 10 buzzbaits times the number of ways to choose 7 out of 9 spinnerbaits. This is calculated as 10 choose 1 times 9 choose 7.

3. If Henry chooses 2 buzzbaits, he must choose 6 spinnerbaits. The number of ways to do this is the number of ways to choose 2 out of 10 buzzbaits times the number of ways to choose 6 out of 9 spinnerbaits. This is calculated as 10 choose 2 times 9 choose 6.

The total number of ways Henry can choose 8 lures with fewer than 3 buzzbaits is the sum of the number of ways for each of these scenarios.

Let's calculate:

1. 9 choose 8 = 9
2. 10 choose 1 * 9 choose 7 = 10 * 36 = 360
3. 10 choose 2 * 9 choose 6 = 45 * 84 = 3780

So, the total number of ways is 9 + 360 + 3780 = 4149.