Mathematics 9709 · Probability & Statistics 1

Arranging r of n (ⁿPᵣ)

${}^nP_r$ means: choose $r$ objects from $n$ , and arrange them.
Order matters, so the first slot has $n$ choices, the next has $n-1$ , and you stop after $r$ slots.
In an exam, the fastest version is usually just the descending product.

Fill r slots, then stop

${}^nP_r=\dfrac{n!}{(n-r)!}$ , but by hand read it as $n(n-1)\cdots$ for $r$ factors.

Worked example

First rung: arrange 2 from 5

Answer

{}^5P_2=5\times4=20

Worked example

Arrange only part of the pool

Make 3-digit numbers from the seven digits 3 to 9, with no repeated digit.

Four of 18 girls sit on a 4-seat sofa, and the oldest girl must get a seat.

Put the oldest girl in one of the 4 seats, then fill the other 3 seats from the remaining 17: $4\times{}^{17}P_3=4\times17\times16\times15=16\,320$ .

Common mistake

{}^nP_r

has exactly

r

factors. For

{}^7P_3

, stop at

7\times6\times5

, not

7\times6

Arranging r of n (ⁿPᵣ)

${}^nP_r$ means: choose $r$ objects from $n$ , and arrange them.
Order matters, so the first slot has $n$ choices, the next has $n-1$ , and you stop after $r$ slots.
In an exam, the fastest version is usually just the descending product.

${}^nP_r=\dfrac{n!}{(n-r)!}$ , but by hand read it as $n(n-1)\cdots$ for $r$ factors.

Worked example

First rung: arrange 2 from 5

Answer

{}^5P_2=5\times4=20

Worked example

Arrange only part of the pool

Make 3-digit numbers from the seven digits 3 to 9, with no repeated digit.

Four of 18 girls sit on a 4-seat sofa, and the oldest girl must get a seat.

Put the oldest girl in one of the 4 seats, then fill the other 3 seats from the remaining 17: $4\times{}^{17}P_3=4\times17\times16\times15=16\,320$ .

Common mistake

{}^nP_r

has exactly

r

factors. For

{}^7P_3

, stop at

7\times6\times5

, not

7\times6