Permutations & Combinations Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

8 formulas · 7 subtopics

^nP_r = \frac{n!}{(n-r)!}

💡 Order matters. Number of ways to arrange r items from n distinct items.

⚠ Using nCr when order matters.

^nC_r = \frac{n!}{r!(n-r)!}

💡 Order does not matter. Selection of r items from n distinct items.

⚠ Forgetting to divide by r! when order doesn't matter.

\frac{n!}{p_1! \cdot p_2! \cdot \ldots \cdot p_k!}

💡 n total items with p_1 identical of type 1, p_2 of type 2, etc.

(n-1)!

💡 Fix one object and arrange the rest. For necklace/bracelet, divide by 2.

⚠ Using n! instead of (n-1)! for circular arrangements.

\text{Identical objects into distinct boxes: } ^{n+r-1}C_{r-1}

💡 n identical items into r distinct groups (each group can be empty).

⚠ Forgetting to adjust when minimum constraint applies (subtract first, then apply).

^nC_r = ^{n-1}C_r + ^{n-1}C_{r-1}

💡 Basis of Pascal's triangle. Useful for recursive counting arguments.

D_n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}

💡 Number of permutations where no element is in its original position.

\text{Check digit-sum (for 3,9) or last digits (for 2,4,5,8)}

💡 Divisibility by 3/9: digit sum divisible. By 4: last 2 digits. By 8: last 3 digits.