In the expansion of (a + b)^n, if the coefficient of a^2b^3 is 10, what is the v

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Question: In the expansion of (a + b)^n, if the coefficient of a^2b^3 is 10, what is the value of n?

Options:

Correct Answer: 6

Solution:

The coefficient of a^2b^3 in (a + b)^n is given by C(n, 3). Setting C(n, 3) = 10 gives n = 6.

In the expansion of (a + b)^n, if the coefficient of a^2b^3 is 10, what is the value of n?

In the expansion of (a + b)^n, if the coefficient of a^2b^3 is 10, what is the value of n?

Step 1: Understand that (a + b)^n is a binomial expansion where 'n' is a positive integer.
Step 2: The general term in the expansion of (a + b)^n is given by C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient.
Step 3: Identify the term we are interested in: a^2b^3. Here, the exponent of 'a' is 2 and the exponent of 'b' is 3.
Step 4: In the term a^(n-k) * b^k, we can set n-k = 2 (for a^2) and k = 3 (for b^3).
Step 5: From k = 3, we know that n - 3 = 2, which means n = 5.
Step 6: The coefficient of a^2b^3 is given by C(n, 3). We need to find C(5, 3).
Step 7: Calculate C(5, 3) = 5! / (3! * (5-3)!) = (5 * 4) / (2 * 1) = 10.
Step 8: Since we found that C(n, 3) = 10 when n = 5, we confirm that the value of n is 5.

Binomial Expansion – Understanding how to find coefficients in the expansion of (a + b)^n using the binomial theorem.
Combinatorics – Applying combinations to determine the coefficients of specific terms in the binomial expansion.