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)
