Permutations (nPr)

${}^nP_r = \dfrac{n!}{(n-r)!}$ counts the ways to choose $r$ items from $n$ and arrange them, selection with order. It’s the package for “how many ways can $r$ of these $n$ things fill $r$ distinct positions”.

8 runners; how many ways can gold, silver and bronze be awarded? ${}^8P_3 = \dfrac{8!}{5!} = 8 \times 7 \times 6 = \mathbf{336}$

Notice the cancellation: ${}^8P_3$ is the slot product $8 \times 7 \times 6$. The formula and the slots are the same object, which means if you ever doubt the formula, draw the slots; if the slots are tedious, use the formula. Both earn the marks.

The order test

Use ${}^nP_r$ when the positions are distinguishable: medals, ranked prefects (chairman/secretary/treasurer), letters arranged into “words”, digits into numbers, books onto a shelf choosing 4 of 9. If swapping two chosen items produces a different outcome, order matters → ${}^nP_r$ (or slots). If a swap changes nothing, a committee, a team, a handful of cards, order doesn’t matter and you want $\binom{n}{r}$. The relationship: ${}^nP_r = \binom{n}{r} \times r!$, choose the set, then arrange it; many harder questions are exactly that two-step in disguise.

Restrictions inside permutations

Same discipline as everywhere in the topic, constrained positions first:

From 9 different books, arrange 4 on a shelf with a particular book at the left end. Anchor it: 1 way. Fill the other three positions from the remaining 8: $8 \times 7 \times 6 = \mathbf{336}$

“Two particular items both selected and adjacent” combines the block method with selection, glue, count units, multiply by internal arrangements. Write each factor with a one-word label (“block”, “internal”, “rest”); labelled factors are followable, followable is markable.

Common mistakes

• ${}^nP_r$ used for committees (order doesn’t matter there)
• $n$ and $r$ swapped in the formula (${}^rP_n$ is meaningless, the big number leads)
• Restrictions applied after free counting
• ${}^nP_r = \binom{n}{r} \times r!$ forgotten, making “choose then arrange” questions look unfamiliar
• Calculator ${}^nP_r$ button trusted blind where the slot logic would have caught a misread

Full topic context: P&C notes · the unordered twin: combinations $\binom{n}{r}$.