answer:Using the principle that the number of elements in the union of two sets is equal to the sum of their individual elements minus the number of elements they share (i.e., n(A ∪ B) = n(A) + n(B) - n(A ∩ B)), we can find n(A ∩ B) with the given information: n(A) = 24, n(B) = 69, and n(A ∪ B) = 81. Applying the formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B) 81 = 24 + 69 - n(A ∩ B) n(A ∩ B) = 24 + 69 - 81 n(A ∩ B) = 12. Therefore, the correct answer is C. The intersection of A and B contains 12 elements.

answer:Using the inequalities in (1) from the original answer, we have frac{1}{left(frac n2log frac n2right)^2}<frac{1}{left(log n!right)^2}<frac{1}{(nlog n)^2} tag 3 Since the series sum_{n=3}^{infty}frac{1}{(nlog n)^2} converges by the integral test, it follows from the squeeze theorem and (3) that the series sum_{n=3}^{infty}frac{1}{log (n!)^2} also converges.

answer:The binomial coefficient displaystyle{n choose k} can be extended to non-integer values, including negative numbers, using the gamma function or by the definition: {alpha choose k} = frac{Gamma(alpha + 1)}{Gamma(k + 1)Gamma(alpha - k + 1)} where Gamma is the gamma function. For alpha = -2 and k = 4, the binomial coefficient is calculated as follows: {-2 choose 4} = frac{Gamma(-2 + 1)}{Gamma(4 + 1)Gamma(-2 - 4 + 1)} = frac{Gamma(-1)}{Gamma(5)Gamma(-5)} Since Gamma(n) is undefined for negative integers, we need to use the reflection formula: Gamma(z)Gamma(1-z) = frac{pi}{sin(pi z)} Applying this, we get: {-2 choose 4} = frac{1}{Gamma(5)frac{pi}{sin(pi(-5))}Gamma(-5)} = frac{sin(5pi)}{piGamma(5)^2} Since sin(5pi) = 0, displaystyle{-2 choose 4} is undefined in this context.

answer:To determine if Beehive Limited should acquire the brand, we can calculate the net present value (NPV) of the investment. NPV = -Initial investment + (Annual cash flows / (1 + Required rate of return)^n) where n is the number of years in the brand's useful life. NPV = -7,000,000 + (1,200,000 / (1 + 0.1)^1) + ... + (1,200,000 / (1 + 0.1)^8) NPV = -7,000,000 + 6,797,447 NPV = -202,553 Since the NPV is negative, Beehive Limited should not acquire the "Honeydew" brand as the investment would not meet the company's required rate of return.