If the Binomial Theorem is applied to (x + y)^4, what is the term containing x^2

If the Binomial Theorem is applied to (x + y)^4, what is the term containing x^2y^2?

If the Binomial Theorem is applied to (x + y)^4, what is the term containing x^2y^2?

Step 1: Identify the expression we are expanding, which is (x + y)^4.
Step 2: Understand that the Binomial Theorem states that (a + b)^n can be expanded into a sum of terms of the form C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient.
Step 3: In our case, a = x, b = y, and n = 4.
Step 4: We want to find the term that contains x^2y^2. This means we need x raised to the power of 2 and y raised to the power of 2.
Step 5: To find the correct term, we need to determine k. Since we want x^2, we have n - k = 2, which means k = 4 - 2 = 2.
Step 6: Now we can use the binomial coefficient C(4, 2) to find the coefficient of the term. C(4, 2) = 4! / (2! * (4-2)!) = 6.
Step 7: The term we are looking for is C(4, 2) * x^2 * y^2, which is 6 * x^2 * y^2.
Step 8: Therefore, the term containing x^2y^2 in the expansion of (x + y)^4 is 6x^2y^2.

Binomial Theorem – The Binomial Theorem provides a formula for expanding expressions of the form (x + y)^n, where n is a non-negative integer.
Combinatorial Coefficients – The coefficients in the expansion are determined by the binomial coefficients, represented as nCk, which count the number of ways to choose k elements from a set of n elements.
Term Identification – Identifying specific terms in the expansion requires understanding how to select the appropriate powers of x and y.