Prove that - k-0 10 20Ck-219-12 20C10

Let S=∑ _ k=0 ^ 10 _ #160; ^ 20 C _ k = _ #160; ^ 20 C _ 0 + _ #160; ^ 20 C _ 1 + _ #160; ^ 20 C _ 2 +....+ _ #160; ^ 20 C _ 10 ....(1) In ...

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Binary Operation

Prove that ...

Question

Prove that
$\sum_{k = 0}^{10}^{20} C_{k} = 2^{19} + \frac{1}{2}$ $^{20} C_{10}$

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^{20} C_{0} +^{20} C_{1} + \dots +^{20} C_{8}

If P= $⎡ ⎢ ⎣ \begin{matrix} x 00 0 y 0 00 z \end{matrix} ⎤ ⎥ ⎦ a n d Q = ⎡ ⎢ ⎣ \begin{matrix} a 00 0 b 0 00 c \end{matrix} ⎤ ⎥ ⎦$ then prove that

$P Q = ⎡ ⎢ ⎣ \begin{matrix} x a 00 0 y b 0 00 z c \end{matrix} ⎤ ⎥ ⎦ = Q P$

r, k, p \in W

\sum r + k + p = 10^{30} C_{r} \cdot^{20} C_{k} \cdot^{10} C_{p}

(k + 1)^{2} . k! = (k + 1) (k + 1)!

n \sum k = 1 k 2^{- k} = 2 [1 - 2^{- n} - n {.2}^{- (n + 1)}]

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