Lecture on Permutations and Combinations

Introduction

• Permutation: Arrangement where the order matters.
  • Example: ABC, CAB are different permutations.
• Combination: Grouping where the order doesn’t matter.
  • Example: ABC, CAB are the same combination.

Examples

1. Three Letters: A, B, C

   • Permutations: Different orders like ABC, CAB. Total permutations = 2.
   • Combinations: Only 1 combination because order doesn’t matter.

2. Four Letters: A, B, C, D (Choose 2)

   • Permutations: List all arrangements (AB, AC, AD, BA, etc.). Total = 12.
   • Combinations: Consider only groupings without regard to order (AB = BA). Total = 6.

Calculating Permutations and Combinations

• Permutation Formula (nPr):

  • Formula: ( \frac{n!}{(n-r)!} )
  • Example: 4P2 for 4 letters choosing 2: ( \frac{4!}{2!} = 12 )

• Combination Formula (nCr):

  • Formula: ( \frac{n!}{(n-r)!r!} )
  • Example: 4C2 for 4 letters choosing 2: ( \frac{4!}{2! \times 2!} = 6 )

Example Problems

1. Arranging 3 Books from 7

   • Order matters: Use permutation, 7P3.
   • Calculation: ( \frac{7!}{4!} = 210 )

2. Arranging 5 Books on a Shelf

   • Fundamental counting principle: 5 × 4 × 3 × 2 × 1 = 120.
   • Permutation: 5P5 = 120.

3. Forming Teams of 4 from 12 Engineers

   • Order doesn't matter: Use combination, 12C4.
   • Calculation: ( \frac{12!}{8! \times 4!} = 495 )

Special Cases

1. Arranging "Alabama"

   • Letters: A repeats 4 times.
   • Formula: ( \frac{7!}{4!} = 210 )

2. Arranging "Mississippi"

   • Letters: I repeats 4 times, S repeats 4 times, P repeats 2 times.
   • Formula: ( \frac{11!}{4! \times 4! \times 2!} = 34,650 )

Conclusion

• Permutations focus on order: Arrangements matter.
• Combinations focus on grouping: Order does not matter.
• Use provided formulas to calculate numbers efficiently.

Additional Notes:

• 0 factorial (0!) is 1, an important point to remember.
• Videos available for further understanding in related mathematical topics.