Q81 / 160 · WBJEE 2012 · Mathematics

The sum $1 \times 1!+2 \times 2!+\ldots+50 \times 50$ ! equals

  • A
    $51 !$
  • Correct answer
    B
    $51 !-1$
  • C
    $51 !+1$
  • D
    $2 \times 51 !$

Solution

$1 \times 1!+2 \times 2!+\ldots+50 \times 50!$ $=(2-1) 1!+(3-1) 2!+(4-1) 3 !$ $+\ldots+(51-1) 50!$ $=(2 !-1!)+(3 !-2!)+(4 !-3 !)+(51!-50!)$ $+\ldots+(511-501)$ $=511-11=511-1$

Hence, the correct answer is option: 'B'.

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