Q.The word 'COMPUTER' has 8 distinct letters. In how many ways can 4 of its letters be selected and arranged to form a new arrangement of 4 letters?

"Selected AND arranged" from a fixed pool of distinct objects is exactly what $^{n}P_{r}$ measures directly.

$^{8}P_{4} = 8\times7\times6\times5 = 1680$.

COMPUTER has letters C, O, M, P, U, T, E, R — 8 letters, all distinct. Selecting 4 of them and arranging those 4 in order is precisely $^{8}P_{4}$ (never computed as a separate select-then-arrange calculation, since $^{n}C_{r}\times r!={}^{n}P_{r}$ already covers this):

Multiplying the four descending factors: $8\times7=56$, $56\times6=336$, $336\times5=1680$.

Via selection then arrangement separately: choose 4 letters out of 8 in $^{8}C_{4} = 70$ ways, then arrange each chosen set of 4 in $4! = 24$ orders: $70\times24 = 1680$ — matching $^{8}P_{4}$ directly, confirming the $^{n}C_{r}\times r! = {}^{n}P_{r}$ identity.

Computing only $^{8}C_{4}=70$ (the number of SETS of 4 letters) and stopping there, forgetting the question also asks for each set to be ARRANGED — the wording "selected and arranged" signals the full permutation count is needed, not just the selection count.