Show that 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 - 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

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Properties of Determinants

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

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\frac{{^{n} C}_{r} + 4 {^{n} C}_{r + 1} + 6 {^{n} C}_{r + 2} + 4 {^{n} C}_{r + 3} + {^{n} C}_{r + 4}}{{^{n} C}_{r} + 3 {^{n} C}_{r + 1} + 3 {^{n} C}_{r + 2} + {^{n} C}_{r + 3}} = \frac{n + k}{r + k}

^{m} C_{r} +^{m} C_{r - 1} .^{n} C_{1} +^{m} C_{r - 2},^{n} C_{2} + . . . +^{m} C_{1} .^{n} C_{r - 1} +^{n} C_{r} =

Δ_{1} = ⎛ ⎜ ⎝ \begin{matrix} a & b & 2 c d & e & 2 f x & y & 2 z \end{matrix} ⎞ ⎟ ⎠ & Δ_{2} = ⎛ ⎜ ⎝ \begin{matrix} f & 2 d & e 2 z & 4 x & 2 y c & 2 a & b \end{matrix} ⎞ ⎟ ⎠

Δ_{2} = π . Δ_{1}

π

n

r

1

x

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

(2 \leq r \leq n)

{^{n} C}_{r} + 2 {^{n} C}_{r + 1} + {^{n} C}_{r + 2}

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