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  4. The sum 1× 1! + 2 × 2! + ... + 50 × 50! equals

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Q. The sum $1\times 1! + 2 \times 2! + ... + 50 \times 50!$ equals

WBJEEWBJEE 2012Permutations and Combinations

Solution:

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