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------------------------------------------------ What is the coefficient of [tex]x^5y^{13}[/tex] in the expansion of [tex](3x - 2y)^{15}[/tex]?

a) -114,688
b) -42,375
c) 42,375
d) 114,688

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:

  1. The term's general form is C(15, k) * (3x)^(15-k) * (-2y)^k.
  2. Given that the power of x in the term x⁵ is 5, so (15 - k) must be 5. Thus, k = 10.
  3. Compute the coefficient: C(15, 10) * (3)^5 * (-2)^10.
  4. Calculating the values gets us C(15, 10) = 3003, (3)^5 = 243, and (-2)^10 = 1024.
  5. Multiplying these together: 3003 * 243 * 1024 gives us the coefficient -42,375.

So, the correct answer is (b) -42,375.