If the sum of the coefficients in the expansion of (x + y)ⁿ is 4096, then the greatest coefficient i

Q: Why do we set $x=1$ and $y=1$ to find the sum of coefficients? Isn't that just a random guess?

It is not a guess, but a fundamental property of polynomials! If you have a polynomial $P(x, y) = \sum a_i x^i y^j$, then $P(1, 1)$ is exactly the sum of all coefficients $a_i$. It is a powerful shortcut that saves you from expanding the entire expression.

Q: What if $n$ was odd? Would there still be one middle term?

Great question! If $n$ is odd, there are $n+1$ terms, which is an even number of terms. This means there are two middle terms, both having the same maximum value. For example, if $n=3$, the terms are $^3C_0, ^3C_1, ^3C_2, ^3C_3$, and the middle coefficients $^3C_1$ and $^3C_2$ are equal.

Q: Is there a faster way to calculate $^{12}C_6$ without expanding everything?

Yes! Always look for cancellations before multiplying. Write out the factorials and cancel common terms in the numerator and denominator first. It reduces the risk of calculation errors significantly compared to calculating the full factorials first.

Question

If the sum of the coefficients in the expansion of $(x + y)^n$ is 4096, then the greatest coefficient in the expansion is .

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler on the path of JEE mastery. Today, we are going to peel back the layers of a classic binomial problem. It might look like a simple algebraic expression, but it holds a beautiful geometric secret.

We start with the expansion of $(x + y)^n$. You know the drill: it is a sum of terms, each with a coefficient $^nC_r$. The problem tells us that the sum of these coefficients is $4096$.

But how do we find the sum of coefficients without actually expanding the whole thing? Here is the secret: the **'Sum of Coefficients' trick**.

If you have any polynomial $P(x, y)$, the sum of its coefficients is simply $P(1, 1)$. When you set $x=1$ and $y=1$, every term $x^{n-r}y^r$ becomes $1^{n-r}1^r$, which is just $1$. The variables vanish, leaving only the coefficients behind.

So, our sum becomes:

$$ (1 + 1)^n = 2^n $$

We are given that this sum is $4096$. Now, we have a simple equation: $2^n = 4096$.

### Unlocking the Power

Now, we need to solve for $n$. We know that $2^{10} = 1024$.

If we multiply by $2$, we get $2^{11} = 2048$. Multiply by $2$ once more, and we arrive at $2^{12} = 4096$.

Comparing the exponents, we find that $n = 12$. We have successfully unlocked the power of our binomial expansion. Our expression is $(x + y)^{12}$.

### The Symmetry of Pascal's Mountain

Imagine standing at the base of a mountain. The binomial coefficients $^nC_0, ^nC_1, \dots, ^nC_n$ form the rows of Pascal's Triangle. They start small at the edges, climb steadily, and reach their peak at the very center.

For an even $n$, like our $n=12$, there is a single, unique peak. This peak is the middle term. The position of this term is $\frac{n}{2} + 1$, which means the greatest coefficient is $^{n}C_{n/2}$.

Substituting $n=12$, we are looking for the value of $^{12}C_6$. This is the tallest bar in our distribution, the summit of our mountain.

### The Final Ascent

Now, we must calculate $^{12}C_6$ using the formula:

$$ ^{12}C_6 = \frac{12!}{6! 6!} $$

This is where many students stumble, not because the math is hard, but because they rush. Let's be precise. We write the numerator as $12 	imes 11 	imes 10 	imes 9 	imes 8 	imes 7 	imes 6!$.

We keep the $6!$ in the numerator to cancel one of the $6!$ terms in the denominator. This leaves us with:

$$ \frac{12 	imes 11 	imes 10 	imes 9 	imes 8 	imes 7}{6 	imes 5 	imes 4 	imes 3 	imes 2 	imes 1} $$

Now, let's cancel carefully. $6 	imes 2 = 12$, which cancels the $12$ on top. $5$ cancels $10$ to leave $2$. $4$ cancels $8$ to leave $2$. $3$ cancels $9$ to leave $3$.

We are left with $11 	imes 2 	imes 3 	imes 2 	imes 7$. Grouping these, we get $11 	imes 12 	imes 7$.

Since $12 	imes 7 = 84$, we calculate $11 	imes 84$, which equals $924$. The summit is reached!

The greatest coefficient is **$924$**. Remember this process: identify the sum, find the power, visualize the symmetry, and calculate with precision. You have mastered this concept.

Animated Solution for Mathematics - Binomial Theorem: If the sum of the coefficients in the expansion of $(x + y)^{n}$ is 4096, then the greatest coefficient in the expansion is .

Enter Numerical Value:

Visualized Solution

Understanding $(x + y)^{n}$

Sum of Coefficients Trick

Equating to Find $n$

Solving for $n$

Visualizing the Coefficients

The Greatest Coefficient

Identifying the Middle Term

Expanding $^{12} C_{6}$

Canceling Factorials

Simplifying the Fraction

Final Multiplication

Conclusion

The Sigma Insight: Properties of Binomial Coefficients

Analyzing the Setup

Unlocking the Power

The Symmetry of Pascal's Mountain

The Final Ascent

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Animated Solution for Mathematics - Binomial Theorem: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the sum of the coefficients in the expansion of (x+y)n is 4096, then the greatest coefficient in the expansion is .