Arrangements & Selections

The final exam form of P&C mixes everything: a question with a selection phase and an arrangement phase, plus a restriction, plus an “at least”. The skill is no longer any single formula, it’s decomposing the question into phases and picking the right tool per phase.

The decision checklist

For each phase, ask in order:

1. Am I choosing or arranging (or both)? Choosing → $\binom{n}{r}$; arranging all → factorial; choosing into ordered positions → ${}^nP_r$ or choose-then-arrange ($\binom{n}{r} \times r!$).
2. Any restriction? Handle it first, anchor the fixed item, glue the block, constrain the slot.
3. “At least / at most”? Complement or short-side cases.
4. Do constraints interact? Then split into mutually exclusive cases, count each, add.

Two worked syntheses

From 5 men and 4 women, a committee of 4 is chosen with at least 2 women. In how many ways? Cases: 2W2M, 3W1M, 4W0M (list first, the list is method) $\binom{4}{2}\cdot\binom{5}{2} + \binom{4}{3}\cdot\binom{5}{1} + \binom{4}{4} = 6\cdot10 + 4\cdot5 + 1 = \mathbf{81}$

How many arrangements of the 6 letters of SUNDAE have all three vowels together? Glue U, A, E into a block: 4 units → $4!$ arrangements; vowels arrange internally in $3!$ $4! \times 3! = \mathbf{144}$. (And “vowels NOT all together” $= 6! - 144 = 576$, the complement again.)

Label every factor (“choose women”, “block internal”), labelled products are followable, and followable working is awardable working.

Sanity checks that catch wrong answers

• A “number of ways” must be a positive integer, fractions mean a misplaced division
• Cases should be mutually exclusive and exhaustive, do your cases overlap? Do they cover everything?
• Complements must subtract from the right total (committee questions: total committees, not total people)
• Cross-check small answers by listing: if the answer is 6, can you write the 6 outcomes?

Common mistakes

• One tool forced onto a two-phase question
• Case lists overlapping (double-counting) or incomplete
• Blocks without internal factors; separations computed directly instead of by complement
• “Ways to choose and arrange” answered with $\binom{n}{r}$ alone
• No labels on factors, even correct answers become unverifiable

Full topic context: P&C notes, and note this is a classically avoided topic, which makes it cheap marks at the A/A* boundary for students who drill it.