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Question Number 2196 by Rasheed Soomro last updated on 08/Nov/15

{(a + b)}^{2} = a^{2} + b^{2} + 2 a b

{(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a

{(a + b + c + d)}^{2} = ?

{(a_{1} + a_{2} + \dots + a_{n})}^{2} = ?

D e r i v e a f o r m u l a o r g i v e a t e c h n i q u e .

Answered by 123456 last updated on 08/Nov/15

{(a_{1} + \cdot \cdot \cdot + a_{n})}^{k} = \sum_{α_{1} \cdot \cdot \cdot α_{n}} (\begin{matrix} k \\ α_{1}, \cdot \cdot \cdot α_{n} \end{matrix}) a_{1}^{α_{1}} \cdot \cdot \cdot a_{n}^{α_{n}}

Σ α_{n} = k, α_{i} \in N, k \in N

(\begin{matrix} α \\ α_{1}, \cdot \cdot \cdot, α_{n} \end{matrix}) = \frac{α!}{α_{1}! \cdot \cdot \cdot α_{n}!}

Commented by 123456 last updated on 08/Nov/15

multinomial coeficients and multinomil

expansion

Commented by Rasheed Soomro last updated on 08/Nov/15

W h a t a r e α_{i} ? F o r e x a m p l e f o r {(a_{1} + a_{2} + a_{3} + a_{4})}^{3} w h a t

a r e α_{i} ?

Commented by 123456 last updated on 08/Nov/15

the possible natural pars such that

α_{1} + \cdot \cdot \cdot + α_{n} = k

α_{i} \in N

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