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Question
Find n, if nP6 : nP3 = 120: 1.
Solution
nP6 : nP3 = 120: 1.
`(""^"n""P"_6)/(""^"n""P"_3) = 120/1`
∴ `(("n"!)/(("n" - 6)!))/(("n"!)/(("n" - 3)!)) = 120/1`
∴ `("n"!)/(("n" - 6)!) × (("n" - 3)!)/("n"!) = 120`
∴ `(("n" - 3)!)/(("n" - 6)!) = 120`
∴ `(("n" - 3)("n" - 4)("n" - 5)("n" - 6)!)/(("n"-6)!) = 120`
∴ (n – 3)(n – 4)(n – 5) = 120
∴ (n – 3)[(n – 3) – 1][(n – 3) – 2] = 6 × 5 × 4
Let n – 3 = x
∴ x(x – 1)(x – 2) = 6(6 – 1)(6 – 2)
∴ x = 6
∴ n – 3 = 6
∴ n = 6 + 3
∴ n = 9
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