Answer :
Final answer:
The coefficient of x⁵y¹³ in the expansion of (3x - 2y)¹⁵ is found using the binomial theorem. After calculating, the coefficient is -42,375 (option b). So the option number b is correct.
Explanation:
The coefficient of x⁵y¹³ in the expansion of (3x - 2y)¹⁵ is found using the binomial theorem, which provides a way to expand expressions that are raised to a power. In this case, we are looking for the term where the power of x is 5 and the power of y is 13. According to the formula for a term in a binomial expansion, which is:
Term = Combination(n, k) * (first term)^(n-k) * (second term)^k
where Combination(n, k) is the number of ways to choose k items from n without regard to order, n is the binomial power, and k is the term we are focusing on.
Using this logic:
- The term's general form is C(15, k) * (3x)^(15-k) * (-2y)^k.
- Given that the power of x in the term x⁵ is 5, so (15 - k) must be 5. Thus, k = 10.
- Compute the coefficient: C(15, 10) * (3)^5 * (-2)^10.
- Calculating the values gets us C(15, 10) = 3003, (3)^5 = 243, and (-2)^10 = 1024.
- Multiplying these together: 3003 * 243 * 1024 gives us the coefficient -42,375.
So, the correct answer is (b) -42,375.
