Lecture on Permutations and Combinations

Introduction

• Topic: Permutations and Combinations
• Example using four cats labeled A, B, C, and D
• Comparison between permutations and combinations

Permutations

• Choosing 3 out of 4 cats

  • One goes home with the lecturer
  • One goes to the niece's house
  • One is used to make gumbo (humorous example)

• Goal: Calculate the number of different ways to choose 3 out of 4 cats (expected result is 24)

• Formula:

  • Permutation Formula: $P(n, r) = \frac{n!}{(n-r)!}$
  • $P(4, 3) = \frac{4!}{(4-3)!} = \frac{4!}{1!} = 24$

Combinations

• Choosing 3 out of 4 cats

  • All three cats are taken home as pets

• Formula:

  • Combination Formula: $C(n, r) = \frac{n!}{r!(n-r)!}$
  • $C(4, 3) = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = 4$

Key Differences

• Formulas: The extra factorial term in the combination formula ($r!$ in the denominator)
• Permutations consider order; combinations do not
• Calculations:
  • Permutations: $24$ different ways
  • Combinations: $4$ different ways

Conclusion

• Hands-on examples demonstrate the difference between permutations and combinations
• Previous video suggested to review factorials for more understanding