April 22nd

Pascal's Triangle by Images


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:
  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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