Q.Find the value of $r$ if $^{6}P_{r} = 30$.

With $n$ fixed, try small whole-number values of $r$ and evaluate $^{6}P_{r}$ until the descending-factors product matches the given value.

$^{6}P_{r}$ is the product of the top $r$ descending factors starting at $6$. Try successive whole-number values of $r$:

• $r=1$: $^{6}P_{1} = 6$ (too small)
• $r=2$: $^{6}P_{2} = 6\times5 = 30$ ✓ matches
• $r=3$: $^{6}P_{3} = 6\times5\times4 = 120$ (too large) …

Guessing $r$ without actually verifying the computed value equals $30$ exactly, or stopping at a value of $r$ that gives a merely close (not exact) product; also forgetting that $r$ must be a whol …