The largest coefficient in the expansion of $(1 + x)^{30}$ is ______ .

The largest coefficient in the expansion of (1 + x)³⁰ is ___ .

NCERT Exemplar · Q25

(1 + x)^{30}

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Concept understanding — Binomial Theorem Expansion

The Problem That Started It All

Imagine you have to expand $(x + y)^2$ . That's easy: $x^2 + 2xy + y^2$ . Now try $(x + y)^3$ : $x^3 + 3x^2y + 3xy^2 + y^3$ . Still manageable.

But what about $(x + y)^{10}$ ? Or $(x + y)^{100}$ ? Multiplying it out term by term would take forever. There has to be a pattern — and there is.

The Binomial Theorem is the shortcut that tells you exactly what each term in the expansion of $(x + y)^n$ looks like, without ever having to multiply.

The Pattern You Already Know

Look at the expansions you already know:

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

Notice three things:

The powers of $x$ decrease from $n$ down to $0$ , while the powers of $y$ increase from $0$ up to $n$ . In every term, the exponents add to $n$ .
The coefficients — 1, 4, 6, 4, 1 for $n=4$ — follow a famous pattern called Pascal's triangle.
The number of terms is always $n+1$ .

Note

Pascal's triangle: each number is the sum of the two numbers directly above it.

       1
      1 1
     1 2 1
    1 3 3 1
   1 4 6 4 1
  1 5 10 10 5 1

The Precise Statement

n

Let's break that down.

The symbol $\sum_{k=0}^{n}$ means "add up terms for $k = 0, 1, 2, \dots, n$ ". For each $k$ , the term is:

$\binom{n}{k}$ — the coefficient (read as "n choose k")
$x^{n-k}$ — $x$ raised to the power $n-k$
$y^k$ — $y$ raised to the power $k$

So for $n=4$ , the terms are:

$k$	$\binom{4}{k}$	$x^{4-k}$	$y^k$	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	$\binom{4}{4}=1$	$x^0$	$y^4$	$1 \cdot y^4$

Add them up: $x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$ . Exactly what we had.

Where Do Those Coefficients Come From?

The binomial coefficient $\binom{n}{k}$ counts how many ways you can choose $k$ items from a set of $n$ items. In the expansion, it counts how many ways you can pick $k$ copies of $y$ (and therefore $n-k$ copies of $x$ ) when multiplying $(x+y)$ by itself $n$ times.

Tip

To compute $\binom{n}{k}$ quickly: start at $n$ and multiply $k$ decreasing numbers, then divide by $k!$ .

Example: $\binom{7}{3} = \frac{7 \cdot 6 \cdot 5}{3 \cdot 2 \cdot 1} = 35$ .

The General Term

The $k$ -th term (starting with $k=0$ ) in the expansion is:

\text{General term: } T_{k+1} = \binom{n}{k} x^{n-k} y^k

This is the most useful part for exams. If someone asks "find the 5th term in $(x+y)^{10}$ ", you set $k=4$ (because $T_{k+1}$ means $k=4$ gives the 5th term) and write:

T_5 = \binom{10}{4} x^{6} y^{4}

What If It's Not Just $x$ and $y$ ? …

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The binomial coefficients $\binom{n}{r}$ are symmetric and increase to a maximum value before decreasing. For an even $n$ , the largest coefficient occurs at the middle term, which is $\binom{n}{n/2}$ . For $(1+x)^{30}$ , $n=30$ , so the largest coefficient is $\boxed{\binom{30}{15}}$ .

When we expand a binomial expression like $(a+b)^n$ , the terms involve coefficients that follow a specific pattern. For $(1+x)^n$ , the expansion is given by the Binomial Theorem:

(1+x)^n = \binom{n}{0} + \binom{n}{1}x + \binom{n}{2}x^2 + \dots + \binom{n}{r}x^r + \dots + \binom{n}{n}x^n

The coefficients in this expansion are the binomial coefficients $\binom{n}{r}$ , where $r$ ranges from $0$ to $n$ . We are looking for the largest among these coefficients.

The binomial coefficients have a characteristic property: they first increase, reach a maximum value (or two equal maximum values), and then decrease. This pattern is also symmetric, meaning $\binom{n}{r} = \binom{n}{n-r}$ . For example, in $(1+x)^4$ , the coefficients are $\binom{4}{0}=1, \binom{4}{1}=4, \binom{4}{2}=6, \binom{4}{3}=4, \binom{4}{4}=1$ . The largest is $\binom{4}{2}=6$ .

To find the largest coefficient, we can examine the ratio of consecutive coefficients, $\frac{T_{r+1}}{T_r}$ , where $T_r = \binom{n}{r-1}x^{r-1}$ and $T_{r+1} = \binom{n}{r}x^r$ . The ratio of the coefficients themselves is $\frac{\binom{n}{r}}{\binom{n}{r-1}}$ .

Let's consider the ratio of the $(r+1)$ -th coefficient to the $r$ -th coefficient:

\frac{\text{Coefficient of } x^r}{\text{Coefficient of } x^{r-1}} = \frac{\binom{n}{r}}{\binom{n}{r-1}}

\frac{\binom{n}{r}}{\binom{n}{r-1}} = \frac{n-r+1}{r}

For the coefficients to be increasing, this ratio must be greater than $1$ :

\frac{n-r+1}{r} > 1

n-r+1 > r

n+1 > 2r

r < \frac{n+1}{2}

For the coefficients to be decreasing, this ratio must be less than $1$ :

\frac{n-r+1}{r} < 1

n-r+1 < r

n+1 < 2r

r > \frac{n+1}{2}

This tells us that the coefficients increase as long as $r < \frac{n+1}{2}$ and decrease when $r > \frac{n+1}{2}$ . The largest coefficient(s) will occur around $r = \frac{n+1}{2}$ .

Now, let's apply this to the given problem:

Identify $n$ :

The given expansion is $(1+x)^{30}$ . Comparing this with $(1+x)^n$ , we have $n=30$ .
Determine the position of the largest coefficient:

Since $n=30$ is an even number, $n+1 = 31$ . …

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Q.The largest coefficient in the expansion of $(1 + x)^{30}$ is ______ .

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