# April 22nd

### Pascal's Triangle by Images

#### Notes

• A derangement is a shuffling of $$n$$ ordered things in which none of them end in their natural position. The number of derangements of $$n$$ objects is denoted $$D_n$$.
• $$D_n = n!(\frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \cdots + (-1)^{-n}\frac{1}{n!} )$$
• $$D_n \approx \frac{n!}{e}$$

### Today's Problems

1. A substitution cipher is made by permuting the letters of the alphabet such that every letter is replaced by a different letter (or at least a version of a substitution cipher). How many different codes can be made this way?
2. Fifty poets write a poem as part of a haiku-a-thon. They then give their poems to someone else for review. How many ways can this be done?
3. Use Pascal's Triangle to expand $${(a + 2b)}^{6}$$ (simplified).
4. Find the coefficient of $$x^{16}y^{3}$$ in the expansion of $${(x+y)}^{19}$$
5. Prove the hockey-stick pattern, by showing the following using Pascal's triangle:
• $$\sum_{k=2}^{6} \binom{k}{2} = \binom{7}{3}$$
• $$\sum_{k=r}^{n} \binom{k}{r} = \binom{n+1}{r+1}$$
6. Find a simple expression for $$\binom{n}{0} + 5\binom{n}{1} + 5^2\binom{n}{2} + \cdots + 5^n \binom{n}{n}$$.
7. Find an expression for $$\sum_{k=1}^{n} k \binom{n}{k}$$.
8. Cheryl's birthday infinite version
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