Question

If $n$ is a positive integer, then find the value of $\left[\begin{array}{ccc}{}^{n+2}{C}_n& {}^{n+3}{C}_{n+1}& {}^{n+4}{C}_{n+2}\\ {}^{n+3}{C}_{n+1}& {}^{n+4}{C}_{n+2}& {}^{n+5}{C}_{n+3}\\ {}^{n+4}{C}_{n+2}& {}^{n+5}{C}_{n+3}& {}^{n+6}{C}_{n+4}\end{array}\right]$.

Answer 1

$\left|\left|\begin{array}{ccc}{}^{n+2}{C}_n& {}^{n+3}{C}_{n+1}& {}^{n+4}{C}_{n+2}\\ {}^{n+3}{C}_{n+1}& {}^{n+4}{C}_{n+2}& {}^{n+5}{C}_{n+3}\\ {}^{n+4}{C}_{n+2}& {}^{n+5}{C}_{n+3}& {}^{n+6}{C}_{n+4}\end{array}\right|\right|$

we know that

${}^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r{\Rightarrow }^{n+1}{C}_r{-}^n{C}_{r-1}{=}^n{C}_r-------1$
Applying $R_3\to R_3-R_2$ and $R_2\to R_2-R_1$

$\Rightarrow \left|\left|\begin{array}{ccc}{}^{n+2}{C}_n& {}^{n+3}{C}_{n+1}& {}^{n+4}{C}_{n+2}\\ {}^{n+2}{C}_{n+1}& {}^{n+3}{C}_{n+2}& {}^{n+4}{C}_{n+3}\\ {}^{n+3}{C}_{n+2}& {}^{n+4}{C}_{n+3}& {}^{n+5}{C}_{n+4}\end{array}\right|\right|$ (from equation - 1)

Applying $R_3\to R_3-R_2$
$\Rightarrow \left|\left|\begin{array}{ccc}{}^{n+2}{C}_n& {}^{n+3}{C}_{n+1}& {}^{n+4}{C}_{n+2}\\ {}^{n+2}{C}_{n+1}& {}^{n+3}{C}_{n+2}& {}^{n+4}{C}_{n+3}\\ {}^{n+2}{C}_{n+2}& {}^{n+3}{C}_{n+3}& {}^{n+4}{C}_{n+4}\end{array}\right|\right|$ (from equation - 1)

$\Rightarrow \left|\left|\begin{array}{ccc}{}^{n+2}{C}_2& {}^{n+3}{C}_2& {}^{n+4}{C}_2\\ n+2& n+3& n+4\\ 1& 1& 1\end{array}\right|\right|$ (${\because }^n{C}_r={}^n{C}_{n-r}$)
Expanding and simplifying gives
$=\left|\right(n+2)-(n+3\left)\right|=1$