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:
color(white)("x")_7P_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 7xx6 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 7xx6xx5 possibilities).
...and finally, continuing on, there is a combined 7xx6xx5xx4 possibilities for the four positions.
Note that permutations (arrangements) are different from combinations (selections).
The permutation
color(white)("XXX")apple, orange, banana, grape
is different from the permutation
color(white)("XXX")banana, orange, apple, grape
(although they are the same combination)