April 20th

The ideas

The problems

  1. Four cats and five mice enter a race. In how many ways can they finish with a mouse placing first, second, and third?
  2. How many permutations of the letters a,b,c,d,e,f,g contain neither the string bge nor the string eaf?
  3. How many numbers with seven distinct digits can be formed using only the digits 2-9?
  4. How many different signals, each consisting of seven flags arranged in a column, can be formed from three identical red flags and four identical blue flags?
  5. A group of eight scientists is composed of five mathematicians and three geologists.
  6. In how many ways can a team of six be chosen from 20 players so as to:
  7. Let \(k\) and \(n\) be natural numbers with \(k \lt n\). Prove, using logic not formulas, that $$ \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$$
  8. In how many ways can 30 identical dolls be placed on seven different shelves?
  9. A florist sells roses in five different colors.
  10. In how many ways can 18 different books be given to Tara, Danny, Shannon, and Mike so that one person has 6 books, one has 2 books, the other two people have 5 books each?

A cat gif