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