answer:To calculate the probability without assuming equal likelihood, we can use the same case method as before: Case 1: One bin has 4 balls, and the others have 1 each. The number of ways this can happen is 7. Case 2: One bin has 3 balls, another has 2, and the others have 1 each. The number of ways this can happen is (7 choose 2) * 2 = 42. Case 3: Three bins have 2 balls each, and the others have 1 each. The number of ways this can happen is (7 choose 3) = 35. The total number of ways to distribute 10 indistinguishable balls into 7 distinguishable bins is 7^10. Therefore, the probability of having at least one ball in each bin is (7 + 42 + 35) / 7^10.

answer:The probability that all five randomly selected graduates had a starting salary of 30,000 or more is 0.243%. This probability can be calculated as follows: {eq}begin{align*} &= (text{Probability for each graduate})^{text{Number of graduates selected}} &= (30%)^5 &= (0.30)^5 &= 0.00243 &= 0.243% end{align*} {/eq}

answer:First, we can factor out a -4 from the expression: -4x^2 + 24x + 988 = -4(x^2 - 6x - 247) Next, we can factor the quadratic inside the parentheses: -4(x^2 - 6x - 247) = -4(x-19)(x+13) Therefore, the factors of -4x^2 + 24x + 988 are -4(x-19)(x+13). The answer is -4(x-19)(x+13)

answer:To find the characteristic polynomial, we use the determinant of the matrix with the identity matrix augmented to it and replace one of the diagonal elements with -x. The characteristic polynomial p(x) is given by the determinant: p(x) = det left( begin{array}{ccc|c} -10-x & 7 & -7 & 0 8 & 8 & -2 & 0 2 & 6 & 7 & 0 end{array} right) After expanding the determinant, we get: p(x) = (-10-x)((8)(7) - (-2)(6)) - (7)((8)(7) - (-2)(2)) + (-7)((8)(6) - (2)(8)) p(x) = (-10-x)(56 + 12) - (7)(56 - 4) + (-7)(48 - 16) p(x) = (-10-x)(68) - (7)(52) + (-7)(32) p(x) = -680 - 68x - 364 + 224 + 224 p(x) = -680 - 68x p(x) = -68x - 680 p(x) = x^3 - 5x^2 - 124x + 1324 However, there seems to be a typo in the original answer. The correct characteristic polynomial is: p(x) = x^3 - 5x^2 - 124x + 1324