# What is the value of 4P4?

Apr 12, 2017

""_4P_4=color(green)(24)

#### Explanation:

""_4P_4 means the number of ways (permutations) of arranging $4$ items from a collection of $4$ items.

There are $4$ choices for the first item,

and for each of these, there are $3$ remaining choices for the second item (or $4 \times 3 = 12$ choices for selecting and aarranging the first $2$ items),

and for each of these, there are $2$ remaining choices for the third item (or $4 \times 3 x 2 = 24$ choices for selecting and arranging the first $3$ items);

and for the fourth item, there is only $1$ item left; giving $4 \times 3 \times 2 \times 1 = 24$ choices for selecting and arranging the $4$ items).

As a formula
color(white)("XXX")""_nP_k=(n!)/((n-k)!)

So (remembering that 0! =1 by definition)
color(white)("XXX")""_4P_4=(4!)/(0!) = (4xx3xx2xx1)/(0!) =24