📘 Combinatorics Notes
1. Factorial (n!)
- n! = n × (n-1) × (n-2) × ... × 1
- 0! = 1 (by definition) 👉 Counts the number of ways to arrange n distinct objects.
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2. Permutations (Order matters)
a) Without Repetition
P(n,r) = n! ÷ (n-r)! 👉 Choose r objects from n, order matters, no repeats allowed.
Example: Seating 3 students out of 10 chairs → order matters.
b) With Repetition
n^r 👉 Each position can be filled in n ways, repeats allowed.
Example: 4-digit PIN code using 0–9 (digits can repeat).
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3. Combinations (Order does not matter)
a) Without Repetition
C(n,r) = n! ÷ ( r! × (n-r)! ) 👉 Choose r objects from n, order doesn’t matter, no repeats.
Example: Picking 3 team members out of 10 students.
b) With Repetition
C(n+r-1, r) = ( (n+r-1)! ) ÷ ( r! × (n-1)! ) 👉 Choose r objects from n, order doesn’t matter, repeats allowed.
Example: Choosing 3 scoops of ice cream from 5 flavors (flavors can repeat).
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4. Key Differences
- Permutation = Order matters
- Without repeat → P(n,r)
- With repeat → n^r
- Combination = Order does not matter
- Without repeat → C(n,r)
- With repeat → C(n+r-1, r)
👉 Trick to remember:
- PIN codes → Permutations (order matters)
- Teams → Combinations (order doesn’t matter)
- Ice cream scoops → Combination with repetition