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JEE Main201912 Apr 2019Evening ShiftMathematicsBinomial TheoremActual

If   20 C 1 + ( 2 2 )     20 C 2 + ( 3 2 )   20 C 3 + . . . . . + 20 2     20 C 20 = A 2 β , then the ordered pair ( A , β ) is equal to

Options

  1. A380 , 19
  2. B( 420 , 18 )
  3. C( 420 , 19 )
  4. D( 380 , 18 )

Correct answer

B. ( 420 , 18 )

Step-by-step solution

( 1 + x ) n =   n C 0 +   n C 1 x +   n C 2 x 2 + . . . . . . +   n C n x n Diff. w. r. t. x ⇒ n ( 1 + x ) n - 1 =   n C 1   +   n C 2 ( 2 x ) + . . . . . +   n C n n ( x ) n - 1 Multiply by x both side ⇒ n x ( 1 + x ) n - 1 =   n C 1 x +   n C 2 2 x 2 + . . . . . +   n C n n x n Diff. w. r. t. x ⇒ n [ ( 1 + x ) n - 1 + ( n - 1 ) x ( 1 + x ) n - 2 ]   =   n C 1   +   n C 2 2 2 X + . . . . . . +   n C n

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