If the expansion of (x + y)^5 is written out, which term corresponds to x^3y^2?

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Question: If the expansion of (x + y)^5 is written out, which term corresponds to x^3y^2?

Options:

Correct Answer: The 4th term

Solution:

In the expansion of (x + y)^5, the term x^3y^2 corresponds to the 4th term, calculated using the formula C(5, 2)x^3y^2.

If the expansion of (x + y)^5 is written out, which term corresponds to x^3y^2?

If the expansion of (x + y)^5 is written out, which term corresponds to x^3y^2?

Step 1: Understand that (x + y)^5 means we are expanding the expression (x + y) multiplied by itself 5 times.
Step 2: Identify the general term in the expansion of (x + y)^n, which is given by the formula C(n, k) * x^(n-k) * y^k, where C(n, k) is the binomial coefficient.
Step 3: In our case, n is 5 (because we have (x + y)^5) and we want the term x^3y^2.
Step 4: Determine the values of k and n-k for the term x^3y^2. Here, x is raised to the power of 3, so n-k = 3, which means k = 5 - 3 = 2.
Step 5: Now we can find the term corresponding to x^3y^2 using k = 2. The term is C(5, 2) * x^3 * y^2.
Step 6: Calculate C(5, 2), which is the number of ways to choose 2 items from 5. This is calculated as 5! / (2!(5-2)!) = 10.
Step 7: Therefore, the term corresponding to x^3y^2 in the expansion is 10 * x^3 * y^2, which is the 4th term in the expansion.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Binomial Coefficients – Using combinations to find the coefficients of terms in the expansion.
Term Identification – Identifying specific terms in the expansion based on their powers.