Combinations (nCr)

$\binom{n}{r} = \dfrac{n!}{r!(n-r)!}$ counts the ways to choose $r$ items from $n$ when order is irrelevant, committees, teams, hands of cards, subsets.

Choose 4 students from 10 for a committee: $\binom{10}{4} = 210$.

It’s ${}^nP_r$ with the arrangements divided out ($\div\, r!$), which is the intuition: choose-and-arrange, then forget the arranging. Useful symmetry: $\binom{n}{r} = \binom{n}{n-r}$, choosing 4 to include is the same as choosing 6 to exclude; $\binom{10}{8}$ computes faster as $\binom{10}{2} = 45$.

Multi-group selections: multiply the choices

The standard exam escalation draws from labelled pools:

A committee of 5 from 6 men and 4 women, with exactly 3 men: (choose the men) $\times$ (choose the women) $= \binom{6}{3} \times \binom{4}{2} = 20 \times 6 = \mathbf{120}$

Each group’s selection is independent → multiply. Conditions like “more men than women” split into cases (3M2W, 4M1W, 5M0W), each a product, added, list the cases before computing; the case list is the method.

”At least” / “at most”: count the complement

At least one woman on the committee of 5 from 6M + 4W: Total $-$ none: $\binom{10}{5} - \binom{6}{5} = 252 - 6 = \mathbf{246}$

One subtraction beats four added cases, fewer computations, fewer chances to omit one. State the structure in words (“total minus all-male committees”); the stated logic earns credit even through an arithmetic slip. “At most one” = (none) + (exactly one), still fewer cases from the small side.

Mixed select-then-arrange questions

The harder parts chain combinations into permutations: “choose 4 of 9 books, then arrange them on a shelf” $= \binom{9}{4} \times 4!$ ($= {}^9P_4$, consistency check available). When a question mixes a choice phase with an ordering phase, write the two factors separately and label them.

Common mistakes

• $\binom{n}{r}$ where positions are distinguishable (that’s ${}^nP_r$ territory, apply the order test)
• Case lists missing a case (write them down first)
• “At least” built case-by-case when the complement was one line
• Group selections added instead of multiplied
• Answers left as $\binom{6}{3}$ symbols when a number was asked

Full topic context: P&C notes · the synthesis: arrangements & selections.