Marvin the fly starts at (0,0). Each step, Marvin moves one unit right or one unit up. He is trying to get his home at (4,6). However, at (2,4) here is a frog that will eat him if he goes through that point. In how many ways can Marvin reach his home?
The total number of ways to get from (0,0) to (4,6) as learned in basic counting is 4 + 6 choose 4. 10 choose 4 = 210.
The number of ways to get from (0,0) to (2,4) is 6 choose 2, and the ways to get from (2,4) to (4,6) is 4 choose 2. Total ways is 15 * 6 = 90
Why calculate that? So now we can know the total number of ways to get their, and the total number of ways to get from the origin to (4,6) that pass through the frog and subtract the two totals.
210 - 90 = 120 total ways Marvin can reach his home.