Verify that $^{7}P_{3} = {}^{7}C_{3} \times 3!$.

Verify that ⁷P₃ = ⁷C₃ × 3!.

Worked Examples · Example 4

^{7}P_{3} = {}^{7}C_{3} \times 3!

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Left side — compute $^{7}P_{3}$ directly:

^{7}P_{3} = 7\times6\times5 = 210

Right side — compute $^{7}C_{3}$ first, then multiply by $3!$ :

^{7}C_{3} = \frac{7!}{3!\,4!} = \frac{7\times6\times5\times4!}{3!\times4!} = \frac{7\times6\times5}{3!} = \frac{210}{6} = 35

$^{7}C_{3}\times3! = 35\times6 = 210$ …