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JEE Main20205 Sep 2020Morning ShiftMathematicsBinomial TheoremActual

The natural number m , for which the coefficient of x in the binomial expansion of x m + 1 x 2 22 is 1540, is

Correct answer

13

Step-by-step solution

The general term in the binomial expansion, T r + 1 = C r 22 x m 22 - r x - 2 r ⇒ T r + 1 = C r 22 x m ( 22 - r ) - 2 r For the coefficient of x 22   m - m   r - 2   r = 1 ⇒ 22   m - 1 = r ( m + 2 ) ⇒ r = 22 m - 1 m + 2 ⇒ r = 22 m + 44 - 45 m + 2 r = 22 - 3 . 3 . 5 m + 2 ⇒ 22 - r = 3 · 3 · 5 m + 2 Since, C r 22 = C 22 - r 22 So possible values of m = 1 , 3 , 7 , 13 , 43 , but C 3 22 = 1540 ⇒ 22 - r = 3 ⇒ m = 13

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