Question

What is the value of `4P4`?

Answer 1

`""_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 `4xx3=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 `4xx3x2=24` choices for selecting and arranging the first `3` items);

and for the fourth item, there is only `1` item left; giving `4xx3xx2xx1=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`