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JEE Main202229 Jun 2022Evening ShiftMathematicsBinomial TheoremActual

Let the coefficients of x - 1 and x - 3 in the expansion of 2 x 1 5 - 1 x 1 5 15 , x > 0 , be m and n respectively. If r is a positive integer such m n 2 = C r . 15 2 r , then the value of r is equal to ______.

Correct answer

0

Step-by-step solution

The r t h term of 2 x 1 5 - 1 x 1 5 15 ⇒ T r + 1 = C r 15 2 x 1 5 15 - r - 1 x 1 5 r For term having x - 1 15 - r 5 - r 5 = - 1 ⇒ r = 10 ⇒ m = C 10 15 · 2 5 · - 1 10 For term having x - 3 15 - 2 r 5 = - 3 ⇒ r = 15 ⇒ n = C 15 15 · 2 0 · - 1 15 ∴ m n 2 = C 10 15 · 2 5 = C 5 15 · 2 5 On comparing with m · n 2 = C r 15 · 2 r , we get r = 5

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