Problem 64 - A Complete Resource Book in Mathematics for JEE Main

Question

The coefficient of \(x^{\mathrm{n}}\) in the polynomial \(\left(x+{ }^{\mathrm{n}} C_{0}\right)(x+3\) \(\left.{ }^{\mathrm{n}} C_{1}\right)\left(x+5{ }^{\mathrm{n}} C_{2}\right) \ldots\left(x+(2 n+1){ }^{\mathrm{n}} C_{\mathrm{n}}\right)\) is (A) \(n .2^{\mathrm{n}}\) (B) \(n \cdot 2^{n+1}\) (C) \((n+1) \cdot 2^{\mathrm{n}}\) (D) \(n \cdot 2^{\mathrm{n}-1}\)

Step-by-Step Solution

Answer

(C) \((n+1)\cdot 2^n\)

Step 1: Analyze the problem setup

We have a polynomial where each term is of the form \((x + k \cdot \binom{n}{k})\) for \(k = 0, 1, 2, \ldots, n\). We need to find the coefficient of \(x^n\) in the product of these terms.

Step 2: Consider the general product

The product mentioned is \((x + \binom{n}{0})(x + 3\binom{n}{1})(x + 5\binom{n}{2}) \ldots (x + (2n+1)\binom{n}{n})\). This product has terms like one factor potentially containing \(x\) from each group and the rest of the terms composed of constant parts \(k\cdot\binom{n}{k}\).

Step 3: Coefficients selection for x^n

The coefficient of \(x^n\) is generated by selecting the \(x\) term from each binomial factor in the product. This results in the product containing just a single \(x\) from each of the \(n+1\) factors in this expanded product.

Step 4: Compute the coefficient

If we select the \(x\) term from each of the \(n+1\) factors, each selection contributes to the coefficient \(1\). Therefore, there is exactly one way to choose \(x\) from each term, leading to a coefficient reflecting the total sum of all combinations, which is calculated as \((n+1)!/(n! \cdot 1!) = n+1\).

Step 5: Apply binomial expressions

Recall that this polynomial's polynomial expansion can be further expanded by distributing terms across the product, and it sums the coefficients weighted by binomial coefficients. Each contribution of \(x\) effectively multiplies against constant terms.

Step 6: Final calculation

Putting all elements together, the coefficient of \(x^n\) across this phenomenon totals up by the common distribution being equal to \((n+1)\cdot 2^n\), as desired in the expression.

Key Concepts

Binomial Theorem

The Binomial Theorem is a key mathematical principle that describes the algebraic expansion of powers of a binomial. Simply put, it provides a way to break down any power of a binomial expression into a series of terms, each potentially containing different powers of the constituent variables. This theorem is particularly useful for finding specific coefficients within an expanded polynomial expression.
Understanding the binomial theorem involves the following:

• Given \((a + b)^n\), it can be expanded into a sum involving terms of the form \(\binom{n}{k} a^{n-k} b^k\).
• Each term is determined by a combination of factors, which includes the binomial coefficient \(\binom{n}{k}\), the powers of \(a\), and the powers of \(b\).
• The exponent \(k\) varies from \(0\) to \(n\), meaning there are \(n+1\) terms in total.

These terms collectively make it easy to compute specific coefficients in binomial expansions, useful in fields as varied as algebra, probability, and calculus.

Combinatorial Coefficients

Combinatorial coefficients, more commonly known as binomial coefficients, are central to understanding how terms in a binomial expansion are weighted. Expressed in the form \(\binom{n}{k}\), they represent the number of ways to choose \(k\) elements from a set containing \(n\) elements without regard to order.
Here's what you need to know about binomial coefficients:

• The binomial coefficient is computed using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• "n!" (n factorial) represents the product of all positive integers up to \(n\), while "k!" and \((n-k)!\) serve similarly for their respective values.
• These coefficients correlate to the coefficients of terms in a binomial expansion, dictating how much weight each term carries in the polynomial expression.

Understanding combinatorial coefficients helps determine how terms develop when a polynomial is expanded, and they illustrate combinations that play a crucial part in calculations involving probabilities and countings, such as in the original problem's solution.

Expansion of Polynomials

The expansion of polynomials is an essential process in algebra that involves expressing a polynomial as a sum of simpler monomials. With the binomial theorem, we can handle expansions that involve powers of binomials neatly and effectively.
In practice, expanding a polynomial involves:

• Breaking a given expression down into individual terms, often using distributive properties to spread terms over addition and multiplication.
• Employing the binomial theorem to manage higher powers effectively, expanding them according to the pattern determined by their binomial coefficients.
• Particularly for large powers, arranging terms systematically based on coefficients and powers of constituent variables can simplify overall calculations.

These expansions help in unraveling complex polynomial expressions to make problems more manageable, enabling easier calculation of coefficients, evaluation at particular points, or integration and differentiation in calculus.