Chapter Summary

For each of the following counting problems, say whether the answer is ${10\choose 4}$, $P(10,4)$, or neither. If you answer is “neither,” say what the answer should be instead.

1. How many shortest lattice paths are there from $(0,0)$ to $(10,4)$?

2. If you have 10 bow ties, and you want to select 4 of them for next week, how many choices do you have?

3. Suppose you have 10 bow ties and you will wear a different one on each of the next 4 days…

Give a counting question where the answer is $8\cdot 3 \cdot 3 \cdot 5$. Give another question where the answer is $8 + 3 + 3 + 5$.

Explain using lattice paths why $\sum_{k=0}^n {n \choose k} = 2^n$.