Question

Number of terms in expansion of \((\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})^{10}+(\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}})^{10}\) is \(\ldots \ldots\) (a) 5 (b) 6 (c) 7 (d) 8

Short answer

The number of terms in the expansion of \((\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})^{10}+(\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}})^{10}\) is \(6\).

Step-by-step solution

Understand the Binomial Theorem

The binomial theorem is used to find the expansion of a binomial expression raised to some power. It states that for any positive integer n, and any real numbers a and b: \((a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\) Where \({n \choose k} = \frac{n!}{k!(n-k)!}\) denotes the binomial coefficient.

Expanding \((\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})^{10}\) using Binomial Theorem

Using the binomial theorem, we can expand the first term as follows: \((\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})^{10} = \sum_{k=0}^{10} {10 \choose k} (\sqrt{\mathrm{x}})^{10-k} (\sqrt{\mathrm{y}})^k\)

Expanding \((\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}})^{10}\) using Binomial Theorem

Similarly, we can expand the second term as follows: \((\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}})^{10} = \sum_{k=0}^{10} {10 \choose k} (\sqrt{\mathrm{x}})^{10-k} (-\sqrt{\mathrm{y}})^k\)

Combine the expansions of Step 2 and Step 3

Now, we need to combine the expansions from Step 2 and Step 3: \((\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})^{10}+(\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}})^{10} = \sum_{k=0}^{10} {10 \choose k} (\sqrt{\mathrm{x}})^{10-k} (\sqrt{\mathrm{y}})^k + \sum_{k=0}^{10} {10 \choose k} (\sqrt{\mathrm{x}})^{10-k} (-\sqrt{\mathrm{y}})^k\) Observe that when k is odd, the terms in the second sum will become negative and cancel out the corresponding terms in the first sum. When k is even, the terms in the second sum will become positive and double the corresponding terms in the first sum. Finally, the combined expression will contain the terms with even k values only. Let's count these terms.

Counting the terms with even k values

The possible even k values are 0, 2, 4, 6, 8, and 10. Therefore, there are 6 terms in the expansion of \((\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}})^{10}+(\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}})^{10}\) with even k values. So, the number of terms in the expansion is 6 which corresponds to option (b).

Key concepts

Polynomial Expansion

The concept of polynomial expansion is crucial in mathematics, especially in algebra. It involves expressing a polynomial raised to a specific power in terms of its individual terms. This is generally achieved using algebraic processes like the Binomial Theorem or other algebraic identities.

In the context of the original exercise, we are dealing with the expansion of expressions like \((\sqrt{x} + \sqrt{y})^{10}\). The base terms \(\sqrt{x}\) and \(\sqrt{y}\) are involved in a polynomial raised to the 10th power.

The process of expanding such expressions involves expressing them as a sum of terms, where each term is a product of coefficients and the base terms raised to powers that add up to the original exponent. This allows complex expressions to be simplified into manageable parts that are easier to handle or analyze. Understanding polynomial expansions can be extremely helpful when simplifying or solving problems related to polynomials.

Binomial Coefficient

A binomial coefficient plays a significant role in polynomial expansions, especially when using the Binomial Theorem. Binomial coefficients are numbers that are found in the expansion of binomials, represented as \({n \choose k}\). This notation originates from combinatorics and represents the number of ways to choose \(k\) items from a set of \(n\) items without regard to the order of selection.

Mathematically, the binomial coefficient is calculated using the formula:\[{n \choose k} = \frac{n!}{k!(n-k)!}\]This formula involves the factorial operation, which multiplies a sequence of descending natural numbers. For example, \(10! = 10 \times 9 \times 8 \times \ldots \times 1\).

In the original exercise, binomial coefficients are used to determine the coefficients of each term in the expansion. It significantly simplifies the process of setting up the terms in the polynomial expansion by providing straightforward coefficients for each combination of \((\sqrt{x})^{10-k}(\sqrt{y})^k\) in our case. This allows us to systematically derive the full expanded form of the polynomial.

Combinatorics

Combinatorics is a field of mathematics that deals with counting, arranging, and finding patterns among sets of elements. It’s very useful in solving problems related to binomial expansions because it provides methods to determine coefficients and count possible combinations of terms efficiently.

In the framework of the given exercise, combinatorics underpins the determining of the number of terms in the polynomial expansion. As seen with the binomial coefficients, combinatorics is essential to count the number of terms that effectively form our expanded polynomial when the expression \((\sqrt{x} + \sqrt{y})^{10} + (\sqrt{x} - \sqrt{y})^{10}\) is simplified.

• Counting even powers only (like 0, 2, 4, 6, 8, 10 in the example) results from an understanding of combinatorial principles.
• Cancellation of odd power terms is directly related to discerning the pattern of how terms interact when both binomials are expanded and combined.

Combinatorics thereby not only aids in polynomial expansions but also enhances logical reasoning and outcomes in various mathematical contexts.