Permutations and Combinations Cheatsheet

Cheat Codes & Shortcuts

• Permutation: Arrangement of items where order matters.
• Combination: Selection of items where order does not matter.
• Permutation Formula: \( P(n, r) = \frac{n!}{(n-r)!} \)
• Combination Formula: \( C(n, r) = \frac{n!}{r! (n-r)!} \)
• Factorial: \( n! = n \times (n-1) \times \ldots \times 1 \)
• Permutations of n with repetition: \( n^r \)
• Combination with repetition: \( C(n+r-1, r) \)
• Special cases: \( P(n, n) = n! \), \( C(n, 0) = 1 \)
• Sum Rule: Total ways = sum of individual ways (when mutually exclusive)
• Product Rule: Total ways = product of individual ways (independent tasks)

Quick Reference Table

Type	Form	Solution / Property
Permutation	\( P(5, 3) \)	\( \frac{5!}{(5-3)!} = 60 \)
Combination	\( C(5, 3) \)	\( \frac{5!}{3! \cdot 2!} = 10 \)
Permutation with Repetition	3 letters from 4 (repetition allowed)	\( 4^3 = 64 \)
Combination with Repetition	Select 3 from 4 with repetition	\( C(4 + 3 - 1, 3) = C(6, 3) = 20 \)
Factorial	\( 5! \)	\( 120 \)

Advice

Permutations and Combinations Quick Tips

• Permutations: Number of ways to arrange \( r \) objects from \( n \) distinct ones.
• Combinations: Number of ways to select \( r \) objects from \( n \) without caring about order.
• Use factorial notation: Simplify before expanding factorials.
• Repetition Rules: Use formulas for repetition carefully, distinguish from no-repetition cases.
• Check conditions: Confirm whether order matters & repetition allowed before choosing formula.

Permutations and Combinations Speed Quiz

Test your speed with 5 questions! You have 30 seconds per question.