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Question
If nPr = 1814400 and nCr = 45, find n+4Cr+3
Solution
nPr = 1814400, nCr = 45
∴ `(""^"n""P"_"r")/(""^"n""C"_"r") = 1814400/45`
∴ `("n"!)/(("n" - "r")!) xx ("r"!("n" - "r")!)/("n"!) = 1814400/45`
∴ r! = 40320
∴ r! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
∴ r! = 8!
∴ r = 8
Also, nCr = 45
∴ nC8 = 45
∴ `("n"!)/(8!("n" - 8)!)` = 45
∴ n(n – 1)(n – 2)(n – 3)(n – 4)(n – 5)(n – 6) (n – 7) = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3
Comparing on both sides, we get
n = 10
∴ n+4Cr+3 = 14C11
= `(14!)/(11!(14 - 11)!)`
= `(14!)/(11!3!)`
= `(14 xx 13 xx 12 xx 11!)/(11! xx 3 xx 2 xx 1)`
= 364
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