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Question

If x is a positive integer, then $capital delta equals open vertical bar table row cell x factorial end cell cell left parenthesis x plus 1 right parenthesis factorial end cell cell left parenthesis x plus 2 right parenthesis factorial end cell row cell left parenthesis x plus 1 right parenthesis factorial end cell cell left parenthesis x plus 2 right parenthesis factorial end cell cell left parenthesis x plus 3 right parenthesis factorial end cell row cell left parenthesis x plus 2 right parenthesis factorial end cell cell left parenthesis x plus 3 right parenthesis factorial end cell cell left parenthesis x plus 4 right parenthesis factorial end cell end table close vertical bar$ is equal to

$2 (x!) (x + 1)!$
$2 (x!) (x + 1)! (x + 2)!$
$2 (x!) (x + 3)!$
None of these

The correct answer is: $2 left parenthesis x factorial right parenthesis left parenthesis x plus 1 right parenthesis factorial left parenthesis x plus 2 right parenthesis factorial$

$capital delta equals open vertical bar table row cell x factorial end cell cell left parenthesis x plus 1 right parenthesis factorial end cell cell left parenthesis x plus 2 right parenthesis factorial end cell row cell left parenthesis x plus 1 right parenthesis factorial end cell cell left parenthesis x plus 2 right parenthesis factorial end cell cell left parenthesis x plus 3 right parenthesis factorial end cell row cell left parenthesis x plus 2 right parenthesis factorial end cell cell left parenthesis x plus 3 right parenthesis factorial end cell cell left parenthesis x plus 4 right parenthesis factorial end cell end table close vertical bar$
= $x factorial left parenthesis x plus 1 right parenthesis factorial left parenthesis x plus 2 right parenthesis factorial open vertical bar table row 1 cell left parenthesis x plus 1 right parenthesis end cell cell left parenthesis x plus 2 right parenthesis left parenthesis x plus 1 right parenthesis end cell row 1 cell left parenthesis x plus 2 right parenthesis end cell cell left parenthesis x plus 3 right parenthesis left parenthesis x plus 2 right parenthesis end cell row 1 cell left parenthesis x plus 3 right parenthesis end cell cell left parenthesis x plus 4 right parenthesis left parenthesis x plus 3 right parenthesis end cell end table close vertical bar$
Applying $R subscript 1 end subscript rightwards arrow R subscript 2 end subscript minus R subscript 1 end subscript comma R subscript 2 end subscript rightwards arrow left parenthesis R subscript 3 end subscript minus R subscript 2 end subscript right parenthesis$ we get
$equals x factorial left parenthesis x plus 1 right parenthesis factorial left parenthesis x plus 2 right parenthesis factorial$ $open vertical bar table row 0 1 cell 2 left parenthesis x plus 2 right parenthesis end cell row 0 1 cell 2 left parenthesis x plus 3 right parenthesis end cell row 1 cell left parenthesis x plus 3 right parenthesis end cell cell left parenthesis x plus 4 right parenthesis left parenthesis x plus 3 right parenthesis end cell end table close vertical bar$
$equals 2 x factorial left parenthesis x plus 1 right parenthesis factorial left parenthesis x plus 2 right parenthesis factorial$ ,(on simplification).
Trick: Put $x equals 1$ and then match the alternate.

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