# How do you evaluate 7P4?

Mar 18, 2017

840

#### Explanation:

color(white)()^7P_4=(7!)/((7-4)!)

$\frac{7 \times 6 \times 5 \times 4 \times \cancel{3 \times 2 \times 1}}{\cancel{3 \times 2 \times 1}}$

$\left(7 \times 6 \times 5 \times 4\right) = 840$

Mar 18, 2017

${\textcolor{w h i t e}{x}}_{7} {P}_{4} = 840$

#### Explanation:

In general
color(white)("XXX")color(white)(x)_nP_k= (n!)/((n-k)!

therefore
color(white)("XXX")color(white)("x")_7P_4=(7!)/(3!)=(7xx6xx5xx4cancel(xx3xx2xx1))/cancel(3xx2xx1)=840

Another way to think about it:

${\textcolor{w h i t e}{\text{x}}}_{7} {P}_{4}$ means the number of ways of arranging $4$ items from a possible selection of $7$

There are $7$ possibilities for the first position.

For each placement in the first position there are $6$ possibilities for the second position. This means there are $7 \times 6$ possibilities for the first 2 positions.

For each placement in the first 2 positions there are $5$ possibilities for the third position (for a combined $7 \times 6 \times 5$ possibilities).

...and finally, continuing on, there is a combined $7 \times 6 \times 5 \times 4$ possibilities for the four positions.

Note that permutations (arrangements) are different from combinations (selections).

The permutation
$\textcolor{w h i t e}{\text{XXX}}$apple, orange, banana, grape
is different from the permutation
$\textcolor{w h i t e}{\text{XXX}}$banana, orange, apple, grape
(although they are the same combination)