If both n and r be greater then 1- find the value of x if - - x C r n-1 C r n-1 C r-1 x-1 C r n C r n C r-1 x-2 C r n-1 C r n-1 C r-1 -0

Apply C_3+C_2 and note that ^ n 1 C _ r 1 +^ n 1 C _ r =^ n C _ r ∴Δ =| ^ x C _ r amp; ^ n+1 C _ r amp; ^ n C _ r ^ x+1 C _ r ...

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If both n a...

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If both $n$ and $r$ be greater then $1$ , find the value of $x$ if
$Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^{x} C_{r} & ^{n - 1} C_{r} & ^{n - 1} C_{r - 1}^{x + 1} C_{r} & ^{n} C_{r} & ^{n} C_{r - 1}^{x + 2} C_{r} & ^{n + 1} C_{r} & ^{n + 1} C_{r - 1} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$

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∣ ∣ ∣ ∣ \begin{matrix} ^{x} C_{r} & ^{x} C_{r + 1} & ^{x} C_{r + 2}^{y} C_{r} & ^{y} C_{r + 1} & ^{y} C_{r + 2}^{z} C_{r} & ^{z} C_{r + 1} & ^{z} C_{r + 2} \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^{x} C_{r} & ^{x + 1} C_{r + 1} & ^{x + 2} C_{r + 2}^{y} C_{r} & ^{y + 1} C_{r + 1} & ^{y + 2} C_{r + 2}^{z} C_{r} & ^{z + 1} C_{r + 1} & ^{z + 2} C_{r + 2} \end{matrix} ∣ ∣ ∣ ∣ ∣

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