Problem 60 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The coefficient of \(x^{7}\) in the expression \((1+x)^{10}+x(1+x)^{9}+x^{2}(1+x)^{8}+\ldots+x^{10}\) is: [Jan. 7, 2020 (II)] (a) 210 (b) 330 (c) 120 (d) 420

Step-by-Step Solution

Answer

The coefficient of \(x^7\) is 330.

Step 1: Understand the Expression

We have a series of expressions in the form \((1+x)^{10-k}\) where \(k\) goes from \(0\) to \(10\). The overall series is \((1+x)^{10} + x(1+x)^9 + x^2(1+x)^8 + \ldots + x^{10}\). We need to find the coefficient of \(x^7\) in this sum.

Step 2: Calculate the Coefficients for Each Term

For each term \(x^k(1+x)^{10-k}\), we expand using the Binomial Theorem: \[(1+x)^{10-k} = \sum_{m=0}^{10-k}\binom{10-k}{m}x^m\]Thus, the entire term becomes \[x^k\sum_{m=0}^{10-k}\binom{10-k}{m}x^m = \sum_{m=0}^{10-k}\binom{10-k}{m}x^{m+k}\]

Step 3: Identify contributions to \(x^7\) in each expansion

For each \(x^k (1+x)^{10-k}\), determine which \(m\) makes \(m+k = 7\). Thus, set \(m = 7-k\). This implies that the coefficient of \(x^7\) from each expansion is \(\binom{10-k}{7-k}\) when \(7-k \geq 0\) and \(7-k \leq 10-k\).

Step 4: Calculate individual coefficients and sum them

Sum up the valid contributions from each term:- For \(k=0\), \(m=7\), coefficient is \(\binom{10}{7} = 120\)- For \(k=1\), \(m=6\), coefficient is \(\binom{9}{6} = 84\)- For \(k=2\), \(m=5\), coefficient is \(\binom{8}{5} = 56\)- For \(k=3\), \(m=4\), coefficient is \(\binom{7}{4} = 35\)- For \(k=4\), \(m=3\), coefficient is \(\binom{6}{3} = 20\)- For \(k=5\), \(m=2\), coefficient is \(\binom{5}{2} = 10\)- For \(k=6\), \(m=1\), coefficient is \(\binom{4}{1} = 4\)- For \(k=7\), \(m=0\), coefficient is \(\binom{3}{0} = 1\)Summing all these values gives 120 + 84 + 56 + 35 + 20 + 10 + 4 + 1 = 330.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful concept in algebra used to expand expressions that are raised to any power. Specifically, it gives us a way to expand \((a + b)^n\) into a sum of terms of the form \(\binom{n}{k}a^{n-k}b^k\). Here, \(\binom{n}{k}\) is known as a binomial coefficient, and it represents the number of combinations of \(n\) items taken \(k\) at a time.
When we apply the Binomial Theorem to an expression like \((1+x)^{10-k}\), we can express it as:

• \(\sum_{m=0}^{10-k}\binom{10-k}{m}x^m\)

This formula reflects the different ways we can expand the binomial expression by considering the exponents of \(x\) and constant 1 in each expansive term.
In solving problems involving polynomial expansion, the Binomial Theorem is a useful tool that simplifies the process of finding specific coefficients of terms within the expanded form.

Polynomial Expansion

Polynomial expansion involves taking an expression like \((1+x)^n\) and expressing it as a sum of terms. For instance, when dealing with repeated multiplications of a polynomial, it's useful to have a systematic way of expanding it into linear terms. This process breaks down complex expressions into simpler parts, giving us individual terms and their coefficients.

In the context of the given problem, \(x^k(1+x)^{10-k}\) is expanded using this principle. The expansion of each such term results in a series of individual components yeilding powers of \(x\). Each component contributes a portion to the overall coefficient of \(x^7\) in the final expression. Calculating these coefficients involves identifying the appropriate binomial sums from these expanded forms.

• Find all components of the expression that contribute towards the desired power
• Determine the individual coefficients for each contribution

This sequential breakdown helps us systematically determine which terms from the expansion contribute to specific powers of \(x\).

Combinatorics

Combinatorics is a field of mathematics concerned with counting, arranging, and selecting items within a specific set framework. Binomial coefficients, such as \(\binom{n}{k}\), arise naturally in combinatorial problems because they represent how many ways we can choose \(k\) elements from a total of \(n\) elements.

In the expansion of polynomial expressions through the Binomial Theorem, combinatorics helps us determine the appropriate coefficients required for each term:

• Identify what terms contribute to a specific target power during the expansion
• Use combinatorial coefficients to calculate these contributions

By applying these coefficients appropriately, you can determine how often each combination occurs within the polynomial’s expanded form. This method ensures that each term in a polynomial expansion is unique and accurately represented. By calculating such contributions and summing them, we obtain the final result for a particular term, such as the coefficient for \(x^7\) in the overall expression.