Thus the answer to the original question is \({13 \choose 2} - 75 = 78 - 75 = 3\text{.}\) This makes sense now that we see it. The only way to ensure that no kid gets more than 4 cookies is to give two kids 4 cookies and one kid 3; there are three choices for which kid that should be. We could have found the answer much quicker through this observation, but the point of the example is to illustrate that PIE works!
in-context