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प्रश्न
Find n and r if nCr–1 : nCr : nCr+1 = 20 : 35 : 42
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उत्तर
nCr–1 : nCr : nCr+1 = 20 : 35 : 42
∴ nCr–1 : nCr = 20 : 35 and
nCr : nCr+1 = 35 : 42
Take, nCr–1 : nCr = 20 : 35
∴ `("n"!)/(("r" - 1)! ("n" - "r" + 1)!) xx ("r"!("n" - "r")!)/("n"!) = 20/35`
∴ `1/(("r" - 1)!("n" - "r" + 1)("n" - "r")!) xx "r"("r" - 1)!("n" - "r")! = 20/35`
∴ `"r"/("n" - "r" + 1) = 20/35`
∴ 35r = 20n – 20r + 20
∴ 55r – 20n = 20
∴ 11r – 4n = 4 ...(1)
Take nCr : nCr+1 = 35 : 42
∴ `("n"!)/("r"!("n" - "r")!) xx (("r" + 1)! ("n" - "r" - 1)!)/("n"!) = 35/42`
∴ `(("r" + 1) xx "r"! xx ("n" - "r" - 1)!)/("r"!("n" - "r")("n" - "r" - 1)!) = 35/42`
∴ `("r" + 1)/("n" - "r") = 5/6`
∴ 6r + 6 = 5n – 5r
∴ 11r – 5n = – 6 ...(2)
Subtracting equation (2) from equation (1), we get,
11r – 4n = 4
11r – 5n = – 6
– + +
∴ n = 10
Putting n = 10 in the equation (1), we get,
11r – 4(10) = 4
∴ 11r – 40 = 4
∴ 11r = 44
∴ r = 4
Hence, n = 10 and r = 4.
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