Question

For \(\mathrm{r}=0,1, \ldots .10\), let \(\mathrm{A}_{\mathrm{r}}, \mathrm{B}_{\mathrm{r}}\) and \(\mathrm{C}_{\mathrm{r}}\) denote, respectively, the coefficient of \(\mathrm{x}^{\mathrm{r}}\) in the expansions of \((1+x)^{10},(1+x)^{20}\) and \((1+x)^{30}\). Then $$ \sum_{\mathrm{r}=1}^{10} \mathrm{~A}_{\mathrm{r}}\left(\mathrm{B}_{10} \mathrm{~B}_{\mathrm{r}}-\mathrm{C}_{10} \mathrm{~A}_{\mathrm{r}}\right) $$ is equal to A) \(B_{10}-C_{10}\) B) \(\mathrm{A}_{10}\left(\mathrm{~B}_{10}^{2}-\mathrm{C}_{10} \mathrm{~A}_{10}\right)\) C) 0 D) \(C_{10}-B_{10}\)

Short answer

C) 0

Step-by-step solution

Identify the Coefficients Ar, Br, and Cr

The coefficients Ar, Br, and Cr are the coefficients of x^r in the expansions of (1+x)^10, (1+x)^20, and (1+x)^30 respectively. Using the binomial theorem, these coefficients can be determined as Ar = C(10,r), Br = C(20,r), and Cr = C(30,r), where C(n,r) denotes the combination of n items taken r at a time.

Simplifying the Summation

The given summation is sum_{r=1}^{10} Ar * (Br * B10 - Cr * Ar). Since we know that Ar = C(10,r), Br = C(20,r), and Cr = C(30,r), we can plug these in to simplify.

Binomial Theorem Properties

Considering binomial coefficients properties, such as symmetry (C(n,k) = C(n,n-k)) and Pascal's rule (C(n,k) + C(n,k-1) = C(n+1,k)), we can analyze and reduce the expression.

Use Pascal's Rule and Symmetry

By applying Pascal's rule to the relations between the terms of the different binomial expansions (such as Br = C(20,r) = C(20,20-r)), we can directly compare terms Ar and Cr to Br in terms of their binomial coefficients.

Calculate B10 and C10

B10 and C10 are the coefficients of x^10 in their respective binomial expansions and can be evaluated directly as B10 = C(20,10) and C10 = C(30,10).

Evaluate the summation

Summation can now be simplified further knowing B10 and C10. For each r in the sum, Ar * Br * B10 will always have C(10,r) as a factor, and so will Ar * Ar * C10, which will simplify the expression greatly.

Derive the final result

Given the cancellations and comparisons using binomial coefficient properties, the expression simplifies across the entire summation range, resulting in a single term or zero.

Compare the final result with the options

Taking the derived final result from the previous step, compare it to the given options A, B, C, and D to find the correct answer.

Key concepts

Binomial Coefficients

Binomial coefficients are the numbers that occur as coefficients in the expansion of the binomial power \( (a+b)^n \). They play a crucial role in combinatorics, probability, and various mathematical fields.

Let's dive into an example: In the expansion of \( (1+x)^{10} \), to find the coefficient of \( x^r \), where \( r \geq 0 \), we use the binomial coefficient denoted as \( C(n, r) \) or \( \binom{n}{r} \) which is calculated using the formula:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n! \) is the factorial of \( n \) and represents the product of all positive integers up to \( n \). These coefficients can be effortlessly found in Pascal's triangle or calculated using the formula above.

To expand a binomial expression, you take each term in the sequence of binomial coefficients and multiply it by the corresponding powers of the variables in the binomial. This is a vital concept in JEE Advanced mathematics due to its wide application in problems involving polynomials and algebraic expansion.

JEE Advanced Mathematics

The Joint Entrance Examination (JEE) Advanced assessment in India places a strong emphasis on testing a student's proficiency in mathematics, among other subjects. Within the mathematics section, topics such as algebra, calculus, trigonometry, and geometry are common—each subject requiring a deep understanding of various mathematical principles including binomial theorem expansion.

Understanding the binomial theorem is not only about memorizing the formula but also about recognizing its patterns and properties, such as the binomial coefficients and their symmetrical nature. Proficiency in utilizing combinations, understanding the properties of binomial coefficients, and effectively simplifying complex algebraic expressions underpin many questions in the JEE Advanced mathematics paper.

Students are often required to apply these concepts to solve problems within a limited time, making it essential to be well-practiced. As a JEE aspirant, mastering these aspects through various exercises can significantly boost your mathematical problem-solving skills, especially when tackled within the framework of binomial expansion.

Combinations

Combinations are selections made by taking some or all objects without considering the order of the objects. In the context of binomial expansion, combinations help us determine the specific coefficients that precede each term in the expansion.

Mathematically, a combination is expressed as \( C(n, r) \) or \( \binom{n}{r} \) mentioned earlier, indicating the number of ways we can choose \( r \) elements from a larger set of \( n \) elements. This is particularly useful in probability and combinatorial problems where order does not matter.

The relationship between binomial coefficients and combinations can be observed when performing the binomial theorem expansion. Each term in the expansion of \( (a + b)^n \) corresponds to one of the combinations from the set of \( n \), taken \( r \) at a time, which provides a structured methodology in determining expansion coefficients. Combinatorial reasoning enhances a student's ability to break down and analyze complex problems in JEE Advanced mathematics, where such concepts are frequently applied.