Problem 51 - A Complete Resource Book in Mathematics for JEE Main

Question

The greatest coefficient in the expansion of \((x+y+z+w)^{15}\) is (A) \(\frac{15 !}{3 !(4 !)^{3}}\) (B) \(\frac{15 !}{(3 !)^{3} 4 !}\) (C) \(\frac{15 !}{2 !(4 !)^{2}}\) (D) none of these

Step-by-Step Solution

Answer

The answer is option (A): \(\frac{15!}{3!(4!)^{3}}\).

Step 1: Understand the Multinomial Expansion Concept

The greatest coefficient in the expansion of \((x+y+z+w)^{15}\) can be found using the multinomial theorem. For any positive integer \(n\), the expansion of \((x_1 + x_2 + x_3 + \, ..., \, x_m)^n\) is given by: \[ \sum \frac{n!}{k_1!k_2! \cdots k_m!} x_1^{k_1}x_2^{k_2} \cdots x_m^{k_m} \] where \(k_1 + k_2 + \cdots + k_m = n\). Our task is to identify \(k_1, k_2, k_3, k_4\) that maximize \(\frac{15!}{k_1!k_2!k_3!k_4!}\).

Step 2: Apply Equability in Coefficients

A symmetric case arises when \(k_1 = k_2 = k_3 = k_4 = k\). For them to add up to 15, set \(4k = 15\), which is not possible since \(k\) should be an integer.

Step 3: Adjust Values to Maximize Coefficient

Since equability doesn't work, check integer combinations around \(k=3\) and \(k=4\) that total 15, starting with \((3, 4, 4, 4)\) and permutations. Hence try \((3, 4, 4, 4)\).

Step 4: Calculate the Coefficient for (3, 4, 4, 4)

Substitute \(k_1 = 3\), \(k_2 = 4\), \(k_3 = 4\), \(k_4 = 4\) into the multinomial coefficient formula to get: \[ \frac{15!}{3!4!4!4!} \] Simplify this calculation.

Step 5: Compare to Given Options

Evaluate the expressions in Options (A), (B), (C) to see if any match \(\frac{15!}{3!(4!)^{3}}\). Option (A) matches exactly.

Key Concepts

Multinomial Expansion

The multinomial expansion is a generalization of the binomial theorem. It is applied to the expression \((x_1 + x_2 + x_3 + \, ..., \, x_m)^n\), involving more variables. This means instead of just two terms, you work with several.
For expansion, the formula used is:- \(\sum \frac{n!}{k_1!k_2! \cdots k_m!} x_1^{k_1}x_2^{k_2} \cdots x_m^{k_m}\)- With the condition that the sum of \(k_1+k_2+\cdots+k_m=n\).
In this context, every group of terms within the parentheses, raised to the given power, contributes a term to the full expansion. The individual term's coefficient relies on a _multinomial coefficient_, calculated using the factorial formula above.
Understanding how each term's powers fit this equation is key in creating and solving problems involving multinomial expansions.

Coefficients in Binomial Expansion

In binomial expansion, when we expand expressions like \((x+y)^n\), the coefficients of the terms are determined by the _binomial coefficients_. These coefficients can be found using the formula:
- \(\frac{n!}{k!(n-k)!}\) where \(k\) represents a specific term within the polynomial.
Each coefficient corresponds to a combination of the powers of \(x\) and \(y\). As a precursor to multinomial expansion, understanding binomial coefficients lays the groundwork for tackling more complex problems with more variables.
The formula denotes the number of combinations of \(n\) items selected \(k\) at a time, which is instrumental when determining how terms blend naturally into multinomial scenarios.
Simply put, the coefficients determine the weight each term carries within the expansion.

Maximizing Coefficients

Finding the maximum coefficient in a multinomial expansion involves determining the arrangement of powers that yield the greatest factorial division. Since the arrangement impacts the resulting coefficient size, strategic selection is essential.
Here are steps to maximize:

• Identify integer values \(k_1, k_2, ..., k_m\) that fit \(k_1 + k_2 + ... + k_m = n\).
• Maximize the factorial division: \(\frac{n!}{k_1!k_2!...k_m!}\).
• Try combinations close in size because equal distribution often maximizes coefficients.

In practice, this means identifying the right integer combination where each value complements others, achieving maximum multiplicative interaction.
In the original problem, solving for \((k_1, k_2, k_3, k_4)\) required trials like \((3, 4, 4, 4)\) to match a multinomial distribution that brought forth the greatest factor.
Utilizing symmetry and optimal integer allocation are key strategies in this pursuit.