Question

If x is a positive integer, then $\left|\begin{array}{ccc}x!& (x+1)!& (x+2)!\\ (x+1)!& (x+2)!& (x+3)!\\ (x+2)!& (x+3)!& (x+4)!\end{array}\right|$ is equal to

• A. $2x!(x+1)!$
• B. $2x!(x+1)!(x+2)!$
• C. $2x!(x+3)!$
• D. $2(x+1)!(x+2)!(x+3)!$

Answer 1

Answer: (B)

$\left|\begin{array}{ccc}x!& (x+1)!& (x+2)!\\ (x+1)!& (x+2)!& (x+3)!\\ (x+2)!& (x+3)!& (x+4)!\end{array}\right|$
= $x!(x+1)!(x+2)!$
$\left|\begin{array}{ccc}1& 1& 1\\ x+1& x+2& \left(x+3\right)\\ \left(x+2\right)\left(x+1\right)& \left(x+2\right)\left(x+3\right)& \left(x+3\right)\left(x+4\right)\end{array}\right|$
= $2x!(x+1)!(x+2)!$