In the expansion of $(1 + a)^{m+n}$, prove that coefficients of $a^m$ and $a^n$ are equal.

**OR**

Find the 13th term in the expansion of $\left(9x - \dfrac{1}{3\sqrt{x}} ight)^{18}$, $x
eq 0$.

Question

In the expansion of $(1 + a)^{m+n}$, prove that coefficients of $a^m$ and $a^n$ are equal.

**OR**

Find the 13th term in the expansion of $\left(9x - \dfrac{1}{3\sqrt{x}}ight)^{18}$, $x 
eq 0$.

Accepted Answer

>

> [!TLDR]
> The coefficients of $a^m$ and $a^n$ in $(1+a)^{m+n}$ are $\binom{m+n}{m}$ and $\binom{m+n}{n}$ respectively, and these binomial coefficients are always equal by symmetry.
>
> The general (i.e. $(r+1)$th) term in the expansion of $(1+a)^{m+n}$ is
> $$T_{r+1} = \binom{m+n}{r}a^r$$
>
> Coefficient of $a^m$ (put $r=m$): $\binom{m+n}{m}$
>
> Coefficient of $a^n$ (put $r=n$): $\binom{m+n}{n}$
>
> Using the identity $\binom{N}{k}=\binom{N}{N-k}$ with $N=m+n$: since $N-m = n$,
> $$\binom{m+n}{m} = \binom{m+n}{n}$$
>
> Hence the coefficients of $a^m$ and $a^n$ are equal.
> …

Power	Expansion
$(x + y)^0$	$1$
$(x + y)^1$	$x + y$
$(x + y)^2$	$x^2 + 2xy + y^2$
$(x + y)^3$	$x^3 + 3x^2y + 3xy^2 + y^3$
$(x + y)^4$	$x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$

$k$	$\binom{4}{k}$	$x^{4-k}$	$y^k$	Term
0	$\binom{4}{0}=1$	$x^4$	$y^0$	$1 \cdot x^4$
1	$\binom{4}{1}=4$	$x^3$	$y^1$	$4x^3y$
2	$\binom{4}{2}=6$	$x^2$	$y^2$	$6x^2y^2$
3	$\binom{4}{3}=4$	$x^1$	$y^3$	$4xy^3$
4	$\binom{4}{4}=1$	$x^0$	$y^4$	$1 \cdot y^4$

Q.In the expansion of $(1 + a)^{m+n}$ , prove that coefficients of $a^m$ and $a^n$ are equal. OR Find the 13th term in the expansion of $\left(9x - \dfrac{1}{3\sqrt{x}}\right)^{18}$ , $x \neq 0$ .

Concept understanding — Binomial Theorem Expansion

The Problem That Started It All

The Pattern You Already Know

The Precise Statement

Where Do Those Coefficients Come From?

The General Term

What If It's Not Just $x$ and $y$ ? …

Q.In the expansion of (1+a)m+n(1 + a)^{m+n}(1+a)m+n, prove that coefficients of ama^mam and ana^nan are equal. OR Find the 13th term in the expansion of (9x−13x)18\left(9x - \dfrac{1}{3\sqrt{x}}\right)^{18}(9x−3x​1​)18, x≠0x \neq 0x=0.

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Q.In the expansion of $(1 + a)^{m+n}$ , prove that coefficients of $a^m$ and $a^n$ are equal. OR Find the 13th term in the expansion of $\left(9x - \dfrac{1}{3\sqrt{x}}\right)^{18}$ , $x \neq 0$ .

Unlock everything free for 14 days