Problem 70 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The coefficient of \(\mathrm{t}^{4}\) in the expansion of \(\left(\frac{1-t^{6}}{1-t}\right)^{3}\) (a) 14 (c) 10 (b) 15 (d) 12

Step-by-Step Solution

Answer

The coefficient of \( t^4 \) is 15.

Step 1: Simplify the Expression

Begin by simplifying the expression \( \left(\frac{1-t^6}{1-t}\right)^3 \). This can be written as \( (1 - t^6)^3(1 - t)^{-3} \). The task now is to find the coefficient of \( t^4 \) in this expanded expression.

Step 2: Expand Using Binomial Theorem

Apply the Binomial theorem to each component:1. Expand \( (1 - t^6)^3 \) as \( \sum_{k=0}^{3} \binom{3}{k} (-1)^k t^{6k} = 1 - 3t^6 + 3t^{12} - t^{18} \).2. Expand \( (1 - t)^{-3} \) as \( \sum_{m=0}^{\infty} \binom{m+2}{2} t^m \). Now we need to find the product that results in the coefficients multiplied together for \( t^4 \).

Step 3: Identify Relevant Terms for Desired Power of t

Since we are looking for the coefficient of \( t^4 \), consider the terms that multiply to give \( t^4 \):- 1 from \( (1-t^6)^3 \) and \( \text{term} = \binom{6}{2} t^4 \) from \( (1-t)^{-3} \).The term \( t^6 \) or higher from \( (1 - t^6)^3 \) cannot contribute to \( t^4 \).

Step 4: Calculate Coefficient

The only contributing term is \(1 \cdot \binom{6}{2} t^4 \), therefore the coefficient is \( \binom{6}{2} = \frac{6 \cdot 5}{2 \cdot 1} = 15 \).

Step 5: Conclusion

The coefficient of \( t^4 \) in the given expansion is 15, which matches option (b) in the question.

Key Concepts

Coefficient determination

Determining the coefficient of a specific term in a polynomial expression is an essential skill in algebra and combinatorial mathematics. In our exercise, the main task was to find the coefficient of the term containing \( t^4 \) when expanding the expression \( \left(\frac{1-t^6}{1-t}\right)^3 \). This process involves breaking each part of the polynomial down and methodically using the binomial theorem to identify the appropriate terms.

Once the expression is simplified to \( (1 - t^6)^3(1 - t)^{-3} \), we apply the binomial theorem separately to each part. Each expansion reveals a series of terms that can potentially contribute to the term \( t^4 \). By focusing on terms fulfilling the condition to form \( t^4 \) from these expanded series, the coefficient is calculated. This involves:

• Analyzing how each power of \( t \) in both expressions can pair up through multiplication to yield \( t^4 \).
• Utilizing combinatorial coefficients from the binomial expansions, which indicate the number of ways to select terms.

In this specific scenario, the key term contributing to the desired power of \( t^4 \) relied solely on the term from \( (1-t)^{-3} \), since higher powers in \( (1 - t^6)^3 \) would not help in forming \( t^4 \). The combinatorial coefficient here computed as \( \binom{6}{2} \), resulted in the coefficient of 15 for \( t^4 \).

Polynomial expansion

Polynomial expansion is the process of expressing a polynomial raised to a power as a sum of terms. When working with expressions like \( (1 - t^6)^3(1 - t)^{-3} \), it involves utilizing the binomial theorem. This theorem provides a formula to expand power series, stating that the expression \( (a + b)^n \) can be expanded as:

\[\sum_{k = 0}^{n} \binom{n}{k} a^{n-k} b^k\]
In our exercise, we applied this theorem:

• To the binom \( (1 - t^6)^3 \) resulting in a series of terms: \( 1 - 3t^6 + 3t^{12} - t^{18} \).
• To the geometric series \( (1 - t)^{-3} \), considering it as a special case of the binomial expansion, expressed as \( \sum_{m=0}^{\infty} \binom{m+2}{2} t^m \).

The challenge lies in identifying and selecting the necessary terms from each expanded series to produce the targeted term, such as \( t^4 \). The process often involves careful pairing of terms from different expansions to ensure the exponents in the product match the term we are solving for. By doing so systematically, we can accurately expand complex polynomial expressions and pinpoint specific terms within them.

Combinatorial mathematics

Combinatorial mathematics often plays a significant role when dealing with polynomial expressions and their expansions. It provides the underlying principles that describe how elements combine and interact, crucial when using binomial coefficients.

In expanding polynomial expressions like \( (1-t^6)^3(1-t)^{-3} \), combinatorial mathematics helps by using binomial coefficients \( \binom{n}{k} \), which denote the number of combinations of \( n \) items taken \( k \) at a time. This application is reflected in:

• Identifying how each term in the polynomial contributes to a particular term in the product series.
• Determining the frequency and manner in which terms can be selected from expansions, necessary for precise coefficient calculations.

Understanding and computing these coefficients is simplified through factorial calculations, as in the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]For our task, combinatorial methods simplified finding the coefficient of \( t^4 \). Recognizing that combinatorial-hint expansions \( \binom{6}{2} \) yield a coefficient of 15 in a concise, efficient manner demonstrates the power of combinatorial mathematics in polynomial expansions.