Question

If $n$ and $p$ be two $+$ive integers such that $n\ge p+2$ and
$D(n,p)\left|\begin{array}{ccc}{}^n{C}_p& {}^n{C}_{p+1}& {}^n{C}_{p+2}\\ {}^{n+1}{C}_p& {}^{n+1}{C}_{p+1}& {}^{n+1}{C}_{p+2}\\ {}^{n+2}{C}_p& {}^{n+3}{C}_{p+1}& {}^{n+2}{C}_{p+2}\end{array}\right|$
then prove that $D(n,p)=\frac{{}^{n+2}{C}_3}{{}^{p+2}{C}_3}D(n-1,p-1)$

Answer 1

We know that $\frac{{}^n{C}_r}{{}^{n-1}{C}_{r-1}}=\frac{n}{r}$
$R_1={\frac{n}{p}}^{n-1}{C}_{p-1}$, ${\frac{n}{p+1}}^{n-r}{C}_{p-1}$, ${\frac{n}{p+2}}^{n-1}{C}_{p+1}$
Similarly we can write
$R_2={\frac{n+1}{p}}^n{C}_{p-1}$, ${\frac{n+1}{p+1}}^n{C}_p$, ${\frac{n+1}{p+1}}^n{C}_{p+1}$
$R_3={\frac{n+2}{p}}^{n+1}{C}_{p-1}$, ${\frac{n+2}{p+1}}^{n+1}{C}_p$, ${\frac{n+2}{p+2}}^{n+1}{C}_{p+1}$
Taking $n$ $n+1$ $n+2$ common from $R_1$, $R_2$ and $R_3$ and $\frac{1}{p}$, $\frac{1}{p+1}$, $\frac{1}{p+2}$ common from $C_1$, $C_2$ and $C_3$
$D(n,p)=\frac{(n+2)(n+1)n}{(p+2)(p+1)p}.D(n-1,p-1)$
$=\frac{{}^{n+2}{C}_3}{{}^{(p+2)}{C}_3}.D(n-1,p-1)$