# April 22nd

#### 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

- 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?
- 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?
- Use Pascal's Triangle to expand \({(a + 2b)}^{6}\) (simplified).
- Find the coefficient of \(x^{16}y^{3}\) in the expansion of \({(x+y)}^{19}\)
- Prove the hockey-stick pattern, by showing the following using Pascal's triangle:
- \( \sum_{k=2}^{6} \binom{k}{2} = \binom{7}{3}\)
- \( \sum_{k=r}^{n} \binom{k}{r} = \binom{n+1}{r+1}\)

- Find a simple expression for \( \binom{n}{0} + 5\binom{n}{1} + 5^2\binom{n}{2} + \cdots + 5^n \binom{n}{n} \).
- Find an expression for \( \sum_{k=1}^{n} k \binom{n}{k} \).
- Cheryl's birthday infinite version

Secret Music Video of the Day