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Differentiation of a Determinant

x Cr x Cr + ...

Question

$∣ ∣ ∣ ∣ \begin{matrix} ^{x} C_{r} & ^{x} C_{r + 1} & ^{x} C_{r + 2}^{y} C_{r} & ^{y} C_{r + 1} & ^{y} C_{r + 2}^{z} C_{r} & ^{z} C_{r + 1} & ^{z} C_{r + 2} \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^{x} C_{r} & ^{x + k} C_{r + k} & ^{x + m} C_{r + m}^{y} C_{r} & ^{y + p} C_{r + p} & ^{y + s} C_{r + s}^{z} C_{r} & ^{z + l} C_{r + l} & ^{z + t} C_{r + t} \end{matrix} ∣ ∣ ∣ ∣ ∣$ .Find $k + m + p + s + l + t$ ?

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Solution

L.H.S $= ∣ ∣ ∣ ∣ \begin{matrix} ^{x} C_{r} & ^{x} C_{r + 1} & ^{x} C_{r + 2}^{y} C_{r} & ^{y} C_{r + 1} & ^{y} C_{r + 2}^{z} C_{r} & ^{z} C_{r + 1} & ^{z} C_{r + 2} \end{matrix} ∣ ∣ ∣ ∣$
Applying $C_{2} \to C_{2} + C_{1}$ and $C_{3} \to C_{3} + C_{2}$
$∣ ∣ ∣ ∣ \begin{matrix} ^{x} C_{r} & ^{x} C_{r + 1} +^{x} C_{r} & ^{x} C_{r + 2} +^{x} C_{r + 1}^{y} C_{r} & ^{y} C_{r + 1} +^{y} C_{r} & ^{y} C_{r + 2} +^{y} C_{r + 1}^{z} C_{r} & ^{z} C_{r + 1} +^{z} C_{r} & ^{z} C_{r + 2} +^{z} C_{r + 1} \end{matrix} ∣ ∣ ∣ ∣$
$∣ ∣ ∣ ∣ ∣ \begin{matrix} ^{x} C_{r} & ^{x + 1} C_{r + 1} & ^{x + 1} C_{r + 2}^{y} C_{r} & ^{y + 1} C_{r + 1} & ^{y + 1} C_{r + 2}^{z} C_{r} & ^{z + 1} C_{r + 1} & ^{z + 1} C_{r + 2} \end{matrix} ∣ ∣ ∣ ∣ ∣$ (By Pascal
triangle rub i.e., $^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$ )
Applying $C_{3} \to C_{3} + C_{2}$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^{x} C_{r} & ^{x + 1} C_{r + 1} & ^{x + 1} C_{r + 1} +^{x + 1} C_{r + 2}^{y} C_{r} & ^{y + 1} C_{r + 1} & ^{y + 1} C_{r + 1} +^{y + 1} C_{r + 2}^{z} C_{r} & ^{z + 1} C_{r + 1} & ^{z + 1} C_{r + 1} +^{z + 1} C_{r + 2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^{x} C_{r} & ^{x + 1} C_{r + 1} & ^{x + 2} C_{r + 2}^{y} C_{r} & ^{y + 1} C_{r + 1} & ^{y + 2} C_{r + 2}^{z} C_{r} & ^{z + 1} C_{r + 1} & ^{z + 2} C_{r + 2} \end{matrix} ∣ ∣ ∣ ∣ ∣$

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