Letter counts: $P\to 3$, $Q\to 2$, $R\to 2$, and $S,T,U,V$ each once (7 distinct letters in total). Split into cases by repetition pattern.

Case 1 — 3 alike, 1 different: only P can supply three copies. Choose the different letter from the 6 remaining distinct letters, then arrange 4 letters with 3 identical:

$1\cdot 6\cdot \frac{4!}{3!}=24$.

Case 2 — 2 alike, 2 alike: pick 2 letters from $\{P,Q,R\}$ (each available in $\ge 2$ copies), then arrange:

${}^3{C}_2\cdot \frac{4!}{2!2!}=3\cdot 6=18$.

Case 3 — 2 alike, 2 different: pick the alike letter from $\{P,Q,R\}$, then two distinct from the remaining 6 letters:

${}^3{C}_1\cdot {}^6{C}_2\cdot \frac{4!}{2!}=3\cdot 15\cdot 12=540$.

Case 4 — all 4 different: choose 4 from 7 distinct letters $\{P,Q,R,S,T,U,V\}$ and arrange:

${}^7{C}_4\cdot 4!=35\cdot 24=840$.

Total $=24+18+540+840=1422$.