Name: Middle Term in Binomial Expansion: Illustration
Uploaded: 2019-07-19
Duration: 3 min 24 s
Description: We have previously studied what Binomial Theorem is. We have learnt its expansion. Let's now take a look at this video to observe the Middle Term in Binomial...

Question

An expression of the form

(a + b + c + d + \dots .)

consisting of sum of many distinct symbols is called a multinomial. Show that

(a + b + c)^{n}

is the sum of all terms of the form

\frac{n!}{p! q! r!} a^{p} b^{q} c^{r}

where

p, q

and

r

range over all possible triples of non negative integer such that

p + q + r = n

.

Accepted Answer

Multinomial Theorem:

For non negative integers $b_{1}, b_{2}, b_{3}, . . ., b_{n}$ such that $\sum_{i = 1}^{k} b_{i} = n$ , multinomial co-efficient is $(\begin{matrix} n \\ b_{1}, b_{2}, \dots, b_{k} \end{matrix}) = \frac{n!}{b_{1}! b_{2}! \dots b_{k}!}$ .

${(x_{1} + x_{2} + \dots + x_{k})}^{n} = \sum_{b_{1} + b_{2} + \dots + b_{k} = n} (\begin{matrix} n \\ b_{1}, b_{2}, b_{3}, \dots, b_{k} \end{matrix}) \prod_{j = 1}^{k} x_{j}^{b_{j}}$

In the given question $k$ $= 3$ and $x_{1} = a, x_{2} = b, x_{3} = c$ and $b_{1} = p, b_{2} = q, b_{3} = r$ .

From the multinomial theorem

$(a + b + c)^{n} = \sum_{p + q + r = n} (\overset{n}{p, q, r}) a^{p} b^{q} c^{r}$

$(a + b + c)^{n} =$ $\sum_{p + q + r = n} \frac{n!}{p! q! r!} a^{p} b^{q} c^{r}$

Hence, all the terms are shown as the summation of $\frac{n!}{p! q! r!} a^{p} b^{q} c^{r}$ form.

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

An expression of the form (a+b+c+d+….) consisting of sum of many distinct symbols is called a multinomial. Show that (a+b+c)n is the sum of all terms of the form n!p!q!r!apbqcr where p, q and r range over all possible triples of non negative integer such that p+q+r=n.

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !