Problem 27 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The number of terms in the expansion of \((1+x)^{101}\left(1+x^{2}-x\right)^{100}\) in powers of \(x\) is: (a) 302 (b) 301 (c) 202 (d) 101

Step-by-Step Solution

Answer

The number of terms in the expansion is 302.

Step 1: Understand the problem

We are given the expression \((1+x)^{101}\left(1+x^{2}-x\right)^{100}\) and we need to find the number of distinct terms in its expansion in powers of \(x\). This will involve considering the individual contributions of terms from \((1+x)^{101}\) and \((1+x^{2}-x)^{100}\).

Step 2: Examine the first term \((1+x)^{101}\)

The binomial expansion of \((1+x)^{101}\) has terms involving powers of \(x\) from \(x^0\) to \(x^{101}\). Thus, it has 102 terms corresponding to \(1, x, x^2, \ldots, x^{101}\).

Step 3: Examine the second term \((1+x^{2}-x)^{100}\)

Here, each term in the expansion involves combinations of \(1\), \(x^2\), and \(-x\). Possible powers of \(x\) range from \(0\) (when only \(1\) is chosen) up to \(2 \times 100 = 200\) (when all \(x^2\) terms are taken). We need to ensure we account for all unique powers generated.

Step 4: Determine distinct power combinations

From the second term, the decrease due to \(-x\) can range from 0 up to 100, as each occurrence subtracts 1 from the power. Thus, the minimum power is \(-100\). Combining, the full range of powers from \((1+x^{2}-x)^{100}\) is from \(-100\) to \(200\).

Step 5: Combine terms from both expansions

When multiplied together, powers of \(x\) from \((1+x)^{101}\) and powers of \(x\) from \((1+x^{2}-x)^{100}\) range from \((-100+0)\) to \((101+200)\). Thus, the complete range of powers achievable is \(-100 + \{ 0, 1, \dots, 301 \} = \{ -100, -99, \dots, 101 \} + \{ 0, 1, \dots, 200 \}\). After combining and removing negative or non-achievable powers, distinct powers range from \(0\) to \(301\).

Step 6: Conclusion on the number of terms

Thus, counting from \(0\) to \(301\) yields 302 distinct terms in the final expansion.

Key Concepts

Polynomial Expansion

Polynomial expansion is a method used in algebra to express a polynomial raised to a power as a sum of terms, each involving different powers of its variables. This process is crucial for breaking down complex algebraic expressions into manageable pieces.
In the exercise at hand, polynomial expansion is done using the expressions

• \((1+x)^{101}\)
and
• \((1+x^{2}-x)^{100}\).

For the first polynomial \((1+x)^{101}\), we use the binomial theorem to expand it into terms involving powers of \(x\) from \(x^0\) to \(x^{101}\). The binomial theorem describes expanding \((a+b)^n\) as \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) are binomial coefficients.
Applying it here, each term is derived based on the combinations of the terms \(a\) and \(b\), resulting in a maximum power based on the exponent.In the second polynomial \((1+x^2-x)^{100}\), we see combinations of \(1\), \(x^2\), and \(-x\).
This expansion results in powers of \(x\) ranging from \(-100\) to \(200\), achieved by selecting combinations of these terms. Together, these expanded polynomials are then multiplied, further complicating the expansion by merging the powers of \(x\) based on their combinations.

Combinatorics

Combinatorics is a mathematical field that deals with counting, arrangement, and combination of objects.
In polynomial expansions like this one, combinatorics helps us determine how many distinct terms emerge when different powers are combined.In the given problem, we need to apply combinatorial principles to find the number of unique terms originating from different powers in \((1+x^2-x)^{100}\).
We analyze combinations of these terms, realizing that powers of \(x\) are constructed by mixing \(x^2\) and \(-x\). Using combinations, we cover all possible ways these terms contribute to the total power of \(x\) in the expanded expression.The combinatorial approach involves observing patterns and calculating potential distinct terms by evaluating all feasible powers.
From

• adding and subtracting combinations,
• considering powers and negative contributions,
• allied with the base powers from \((1+x)^{101}\),

we achieve an exhaustive count of each unique term's possibility.
Understanding combinatorics facilitates identifying these terms without manually listing every potential variation, saving time and effort.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operation symbols that combine to represent quantities.
The exercise you're dealing with works extensively with algebraic expressions, mixing and expanding them to reveal various components.Understanding these expressions is central to recognizing how polynomial powers are configured and combined in expansions.
In the given problem, you start with expressions \((1+x)^{101}\) and \((1+x^2-x)^{100}\), focusing on how they separately and collectively alter when expanded.Multiply these polynomials involves algebraic manipulation where you follow rules of arithmetic to develop each term from these expressions,providing a full landscape for various powers of \(x\).
Each operation leads to unique algebraic combinations, showcasing the beauty of algebra in turning complex equations into comprehensive expansions.In algebra, simplifying complex expressions step-by-step and observing how each term contributes vastly enhances your problem-solving skills.
Hence, recognizing each algebraic term, its role and influence within an expression as profound as thisprovides a deep understanding of why and how each result, like the number of terms, appears.