Problem 15 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The coefficient of \(x^{10}\) in the expansion of \((1+x)^{2}\left(1+x^{2}\right)^{3}\) \(\left(1+x^{3}\right)^{4}\) is equal to [Online April \(\mathbf{1 5}, \mathbf{2 0 1 8}\) ] (a) 52 (b) 44 (c) 50 (d) 56

Step-by-Step Solution

Answer

The coefficient of \(x^{10}\) is 36.

Step 1: Expand Each Binomial Term

To find the coefficient of \(x^{10}\) in \((1+x)^{2}(1+x^{2})^{3}(1+x^{3})^{4}\), first consider the expansion of each term separately. The binomial expansion of each term is:1. \((1+x)^{2} = \sum_{k=0}^{2} \binom{2}{k} x^{k}\)2. \((1+x^{2})^{3} = \sum_{m=0}^{3} \binom{3}{m} x^{2m}\)3. \((1+x^{3})^{4} = \sum_{n=0}^{4} \binom{4}{n} x^{3n}\)

Step 2: Determine Required Combinations

The coefficient of \(x^{10}\) is found by adding powers of \(x\) from the expansions such that their total is 10. Let these powers be \(k\), \(2m\), and \(3n\). Solve for non-negative integers with:\[ k + 2m + 3n = 10 \]where \(k \leq 2\), \(m \leq 3\), and \(n \leq 4\).

Step 3: Calculate Each Valid Combination

Find valid \((k, m, n)\) combinations:1. \((k, m, n) = (2, 2, 2)\) gives \(2 + 2 \times 2 + 3 \times 2 = 10\)2. \((k, m, n) = (2, 0, 3)\) gives \(2 + 2 \times 0 + 3 \times 3 = 11\) - invalid3. \((k, m, n) = (0, 2, 2)\) gives \(0 + 2 \times 2 + 3 \times 2 = 10\)4. \((k, m, n) = (0, 4, 0)\) gives \(0 + 2 \times 4 + 3 \times 0 = 8\) - invalidThus, valid combinations are \((2, 2, 2)\) and \((0, 2, 2)\).

Step 4: Calculate Coefficient Sum

For each valid combination, calculate the coefficient:1. For \((2, 2, 2)\): \(\binom{2}{2} \cdot \binom{3}{2} \cdot \binom{4}{2} = 1 \cdot 3 \cdot 6 = 18\)2. For \((0, 2, 2)\): \(\binom{2}{0} \cdot \binom{3}{2} \cdot \binom{4}{2} = 1 \cdot 3 \cdot 6 = 18\)Add coefficients of both valid combinations: \(18 + 18 = 36\)

Key Concepts

Binomial Expansion

Binomial expansion is a method used to expand expressions that are raised to a power. It involves using the binomial theorem to break down expressions of the form \((a + b)^n\) into a sum involving multiple terms. Each term is composed of coefficients and different powers of the two variables, \(a\) and \(b\).

• By using the binomial theorem, any binomial raised to a power \(n\) can be expressed as a sum of terms, where each term is a product of a coefficient and powers of the terms in the binomial.
• The coefficients are determined by combinatorial coefficients, denoted as \(\binom{n}{k}\), which calculate the number of ways to choose \(k\) items from a total of \(n\) items.
• In mathematical notation: \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\).

In the original problem, the concept of binomial expansion allows us to expand each of the three terms: \((1+x)^2\), \((1+x^2)^3\), and \((1+x^3)^4\). These expanded forms are then combined to solve for specific terms in the polynomial expansion.

Combinatorial Coefficients

Combinatorial coefficients, often referred to as "binomial coefficients", play a crucial role in the expansion of binomials. They are represented by \(\binom{n}{k}\) and are calculated as the number of ways to choose \(k\) elements from a set containing \(n\) elements, without regard to the order of elements.

• The formula for calculating combinatorial coefficients is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n!\) denotes factorial, which is the product of all positive integers up to \(n\).
• These coefficients determine the weights of each term in a binomial expansion and are crucial in finding exact terms in larger expressions.
• In our problem, these are used to find coefficients of individual terms in the form: \(\binom{2}{k}\), \(\binom{3}{m}\), and \(\binom{4}{n}\) corresponding to \((1+x)^2\), \((1+x^2)^3\), and \((1+x^3)^4\) respectively.

Understanding these coefficients helps in accurately transferring parts of the expansions that affect the computation of certain desired terms, such as finding particular coefficients in a polynomial.

Polynomial Expansion

Polynomial expansion involves the process of expressing a mathematical expression as a sum of multiple terms. This process requires using binomial expansion and combining separate expansions into a more comprehensive expression.

• Once each binomial term, like \((1+x)^2\), has been expanded using the binomial theorem, we can combine them to achieve a final polynomial.
• Each term in the polynomial corresponds to a specific power of \(x\) with a certain coefficient, calculated using combinatorial coefficients.
• For the exercise in question, we are looking to find the coefficient of \(x^{10}\) from the complete polynomial expansion of the original expression: \((1+x)^2(1+x^2)^3(1+x^3)^4\).

This requires considering all combinations of powers from the expanded forms that could sum to 10. The valid combinations, and their coefficients, determine the coefficient of \(x^{10}\) in the overall polynomial expansion.