Question

$\left|\begin{array}{ccc}10!& 11!& 12!\\ 11!& 12!& 13!\\ 12!& 13!& 14!\end{array}\right|=$ __________.

• A. $-2(10!.11!.12!)$
• B. $2(10!.11!.12!)$
• C. $2(10!.13!)$
• D. $2(10!.12!.13!)$

Answer 1

The correct option is B $2(10!.11!.12!)$

Given $\left|\begin{array}{ccc}10!& 11!& 12!\\ 11!& 12!& 13!\\ 12!& 13!& 14!\end{array}\right|$

$=10!11!12!\left|\begin{array}{ccc}1& 1& 1\\ 11& 12& 13\\ \left(11\right)\left(12\right)& \left(12\right)\left(13\right)& \left(13\right)\left(14\right)\end{array}\right|$

$=10!11!12!\left[\right(12\left)\right(13\left)\right(14)-(12\left)\right(13\left)\right(13)+(11\left)\right(12\left)\right(13)-(11\left)\right(13\left)\right(14)+(11\left)\right(12\left)\right(13)-(11\left)\right(12\left)\right(12\left)\right]$

$=10!11!12!\left[\right(12\left)\right(13\left)\right(14-13)+(11\left)\right(13\left)\right(12-14)+(11\left)\right(12\left)\right(13-12\left)\right]$

$=10!11!12!\left[\right(12\left)\right(13)-2(11\left)\right(13)+(11\left)\right(12\left)\right]$

$=10!11!12!\left[\right(12\left)\right(13)-(11\left)\right(13)+(11\left)\right(12)-(11\left)\right(13\left)\right]$

$=10!11!12!\left[\right(13\left)\right(12-11)+(11\left)\right(12-13\left)\right]$

$=10!11!12![13-11]$

$=2(10!11!12!)$