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What is the value of $4 P 4$ $4P4$ ?

1 Answer

Apr 12, 2017

$_{4} P_{4} = 24$ $""_4P_4=color(green)(24)$

Explanation:

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

There are $4$ $4$ choices for the first item,

and for each of these, there are $3$ $3$ remaining choices for the second item (or $4 \times 3 = 12$ $4xx3=12$ choices for selecting and aarranging the first $2$ $2$ items),

and for each of these, there are $2$ $2$ remaining choices for the third item (or $4 \times 3 x 2 = 24$ $4xx3x2=24$ choices for selecting and arranging the first $3$ $3$ items);

and for the fourth item, there is only $1$ $1$ item left; giving $4 \times 3 \times 2 \times 1 = 24$ $4xx3xx2xx1=24$ choices for selecting and arranging the $4$ $4$ items).

As a formula
$XXX_{n} P_{k} = \frac{n!}{(n - k)!}$ $color(white)("XXX")""_nP_k=(n!)/((n-k)!)$

So (remembering that $0! = 1$ $0! =1$ by definition)
$XXX_{4} P_{4} = \frac{4!}{0!} = \frac{4 \times 3 \times 2 \times 1}{0!} = 24$ $color(white)("XXX")""_4P_4=(4!)/(0!) = (4xx3xx2xx1)/(0!) =24$

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