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

Q: Wait, why do we substitute a = 1 and b = 1 to find the sum of coefficients?

Think of any polynomial $P(a, b) = c_{0}a^{n} + c_{1}a^{n-1}b + \dots + c_{n}b^{n}$. If you set $a = 1$ and $b = 1$, the expression becomes $P(1, 1) = c_{0}(1)^{n} + c_{1}(1)^{n-1}(1) + \dots + c_{n}(1)^{n}$, which is simply the sum of all coefficients $c_{0} + c_{1} + \dots + c_{n}$. It is a powerful, universal shortcut!

Q: Why is the middle term the greatest coefficient?

Binomial coefficients $^{n}C_{r}$ are symmetric, meaning $^{n}C_{r} = ^{n}C_{n-r}$. They increase until they reach the middle and then decrease. For an even n, the peak occurs exactly at the middle index $r = \frac{n}{2}$. It is the point of maximum balance in the distribution.

Q: What if n was odd? Would there still be one greatest coefficient?

Great question! If n is odd, there is no single middle term. Instead, there are two middle terms at $r = \frac{n-1}{2}$ and $r = \frac{n+1}{2}$. Because of symmetry, these two coefficients are equal and both are the greatest.

Question

If the sum of the coefficients in the expansion of $(a + b)^n$ is 4096, then the greatest coefficient in the expansion is

Options:

(A) 1594

(B) 792

(C) 924

(D) 2924

Sigma Percentile · Accepted Answer

### The Mystery of the Binomial Coefficients

Welcome, fellow traveler of the mathematical landscape. Today, we are going to unravel a classic puzzle from the Binomial Theorem.

Imagine you are staring at the expansion of $(a + b)^{n}$. It looks simple enough, but hidden within its structure is a beautiful, symmetric architecture of numbers.

We are told that the sum of all coefficients in this expansion is $4096$. Our mission is to find the absolute greatest coefficient among them. Let us embark on this journey together.

### Unlocking the Exponent $n$

The first step is to identify our exponent, $n$. We have a powerful tool in our arsenal: the Sum of Coefficients Property.

To find the sum of coefficients of any polynomial, we simply set all variables to $1$. When we substitute $a = 1$ and $b = 1$ into our expression $(a + b)^{n}$, it transforms into $(1 + 1)^{n}$, which is $2^{n}$.

We are given that this sum is $4096$. So, we set up the equation:

$$2^{n} = 4096$$

Now, we just need to find the power of $2$ that equals $4096$. We know $2^{10} = 1024$, $2^{11} = 2048$, and doubling that once more gives us $2^{12} = 4096$.

Thus, our exponent is **$n = 12$**.

### The Symmetry of Pascal's Triangle

Now that we know $n = 12$, we are looking for the greatest coefficient in the expansion of $(a + b)^{12}$. Binomial coefficients, which are the values $^{n}C_{r}$, are not random.

They are the entries of Pascal's Triangle. They start small at the edges, rise steadily to a peak in the middle, and then fall back down.

This symmetry is one of the most elegant features of algebra. Since $12$ is an even number, there is exactly one middle term that sits at the very peak of this distribution.

This peak occurs at $r = \frac{n}{2}$. Therefore, the greatest coefficient is $^{12}C_{\frac{12}{2}}$, which is $^{12}C_{6}$.

### The Arithmetic Journey

Now, we must calculate the value of $^{12}C_{6}$. We use the standard combination formula:

$$^{n}C_{r} = \frac{n!}{r!(n-r)!}$$

Substituting our values, we get:

$$^{12}C_{6} = \frac{12!}{6! \cdot 6!}$$

Let us expand this carefully. The numerator is $12 	imes 11 	imes 10 	imes 9 	imes 8 	imes 7 	imes 6!$. We can cancel the $6!$ in the numerator with 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 us simplify this fraction with precision. We see that $6 	imes 2 = 12$, which cancels the $12$ in the numerator. Then, $5$ divides $10$ to leave $2$, $4$ divides $8$ to leave $2$, and $3$ divides $9$ to leave $3$.

We are left with $11 	imes 2 	imes 3 	imes 2 	imes 7$. Multiplying these, we get $22 	imes 6 	imes 7$, which is $22 	imes 42 = 924$.

### Conclusion

And there we have it! The greatest coefficient is **$924$**.

This result is not just a number; it is the culmination of understanding the symmetry of binomial expansion and the power of the sum-of-coefficients property. I hope you can see the beauty in how these numbers align.

Keep practicing, keep questioning, and most importantly, keep falling in love with the logic behind the math.

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

Select Answer:

Visualized Solution

Understanding the Expansion $(a + b)^{n}$

The Sum of Coefficients Property

Setting up the Equation for $n$

Solving for $n$ using Powers of $2$

Symmetry and the Greatest Coefficient

Substituting $n = 12$

Applying the Combination Formula

Expanding the Factorials

Simplifying the Fraction

The Final Answer

The Sigma Insight: Properties of Binomial Coefficients

The Mystery of the Binomial Coefficients

Unlocking the Exponent $n$

The Symmetry of Pascal's Triangle

The Arithmetic Journey

Conclusion

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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 (a+b)n is 4096, then the greatest coefficient in the expansion is

Select Answer:

Visualized Solution

Understanding the Expansion .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (a+b)n

The Sum of Coefficients Property

Setting up the Equation for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n using Powers of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2