Lecture on Permutations and Combinations

Definitions

• Permutation: Arrangement of items where the order matters.
• Combination: Grouping of items where the order does not matter.

Key Concepts

• For permutations, different orders count as different permutations.
• For combinations, different orders of the same items count as the same combination.

Examples

1. Three Letters (A, B, C):

   • Permutations:
     • ABC
     • CAB
     • Total Permutations = 6
   • Combinations:
     • ABC (CAB is considered the same combination)
     • Total Combinations = 1

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

   • List of Permutations:
     • AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC
     • Total Permutations = 12
   • List of Combinations:
     • AB, AC, AD, BC, BD, CD
     • Total Combinations = 6

Formulas

1. Permutations (nPr):

   • Formula: ( nPr = \frac{n!}{(n-r)!} )
   • Example: For 4P2 (choosing 2 from 4)
     • ( 4! / 2! = 4 \times 3 = 12 )

2. Combinations (nCr):

   • Formula: ( nCr = \frac{n!}{(n-r)! \cdot r!} )
   • Example: For 4C2
     • ( 4! / (2! \times 2!) = 6 )

Problem Solving

1. Arranging 3 Books from 7:

   • Determine: Permutation or Combination?
   • Solution:
     • 7P3 = ( \frac{7!}{(7-3)!} = 7 \times 6 \times 5 = 210 )

2. Arranging 5 Books on a Shelf:

   • Use 5P5 = ( 5! = 120 )
   • Alternatively, use the multiplication principle: 5 \times 4 \times 3 \times 2 \times 1 = 120

3. Forming Teams of 4 from 12 Engineers:

   • Solution: Use 12C4
   • Formula application:
     • ( \frac{12!}{(12-4)! \times 4!} = 495 )

4. Arranging Letters in "Alabama":

   • Formula: ( \frac{7!}{4!} = 210 )

5. Arranging Letters in "Mississippi":

   • Formula: ( \frac{11!}{4! \times 4! \times 2!} = 34,653 )

Conclusion

• Permutation: Used when order matters.
• Combination: Used when order does not matter.
• Utilize factorial calculations for solving permutation and combination problems.
• Common application problems involve arranging books, forming teams, and arranging letters in words.