Permutations ($^nP_r$)

A permutation is an arrangement of objects where order matters. When we select $r$ objects from $n$ distinct objects and arrange them in a row, we use:

Why does this formula work? We fill $r$ positions one at a time. The first position can be filled in $n$ ways, the second in $n-1$ ways (since one object is used), and so on until the $r$-th position has $n-r+1$ choices. This gives:

The $(n-r)!$ in the denominator exactly cancels the $n-r$ factorial terms we did not use.

Example: Three-letter arrangements from A, B, C, D, E (no repeats):

$^5P_3 = \dfrac{5!}{(5-3)!} = \dfrac{5!}{2!} = \dfrac{120}{2} = 60$

So there are 60 arrangements. Check your estimate from card 01!

Permutation: ordered arrangement of r objects from n — use when order matters; ^nP_r = n!{(n-r)!} — or equivalently n(n-1)(n-2)(n-r+1)

Pause — copy the permutation formula into your book: $^nP_r = \frac{n!}{(n-r)!} = n(n-1)(n-2)\cdots(n-r+1)$ — $r$ descending factors starting from $n$.

We just saw that $^nP_r = \frac{n!}{(n-r)!}$ counts ordered arrangements of $r$ objects from $n$. That raises a question: the formula gives the same answer whether you use it as a fraction or as $r$ descending factors — which form is faster to compute by hand? This card answers it → write $r$ descending factors starting at $n$: $n \times (n-1) \times \cdots \times (n-r+1)$, avoiding the need to compute large factorials.

You can evaluate $^nP_r$ in two equivalent ways:

• Using the formula: $^nP_r = \dfrac{n!}{(n-r)!}$. Calculate numerator and denominator separately, then divide.
• Using the product: $^nP_r = n \times (n-1) \times (n-2) \times \cdots \times (n-r+1)$, writing exactly $r$ factors starting from $n$.

The product form is often faster for calculation without a calculator. For example:

$^7P_4 = 7 \times 6 \times 5 \times 4 = 840$   (four factors, starting at 7, decreasing by 1 each time)

Check: $\dfrac{7!}{3!} = \dfrac{5040}{6} = 840$ ✓

Product form: write r descending factors starting from n:   n, n-1, n-2, , n-r+1; Formula form: ^nP_r = n!{(n-r)!}

Pause — copy both computation forms into your book: product form — write $r$ descending factors from $n$; formula form — $^nP_r = n!/(n-r)!$; use product form when $r$ is small to avoid computing large factorials.