answer:Let's define the following events: - (A): Machine (alpha) needs attention within one hour. - (B): Machine (beta) needs attention within one hour. - (C): Machine (gamma) needs attention within one hour. The probabilities given are: [ P(A) = 0.9, quad P(B) = 0.8, quad P(C) = 0.85 ] Since ( bar{A}, bar{B}, bar{C} ) represent the probabilities that the corresponding machines do not need attention, we have: [ P(bar{A}) = 1 - P(A) = 0.1, quad P(bar{B}) = 1 - P(B) = 0.2, quad P(bar{C}) = 1 - P(C) = 0.15 ] (1) The probability that none of the machines need attention: We need to find the probability of the event ( bar{A} cap bar{B} cap bar{C} ). Since the events are independent: [ P(bar{A} cap bar{B} cap bar{C}) = P(bar{A}) cdot P(bar{B}) cdot P(bar{C}) ] Substituting the values: [ P(bar{A} cap bar{B} cap bar{C}) = 0.1 cdot 0.2 cdot 0.15 = 0.003 ] Thus, the probability that none of the machines need attention is: [ P(text{None need attention}) = 0.003 ] (2) The probability that at least one machine needs attention: Let ( D ) be the event that at least one machine needs attention. By the complement rule: [ P(D) = 1 - P(bar{A} cap bar{B} cap bar{C}) ] Substituting the value from (1): [ P(D) = 1 - 0.003 = 0.997 ] Thus, the probability that at least one machine needs attention is: [ P(text{At least one needs attention}) = 0.997 ] # Conclusion: The final answers are: 1. The probability that none of the machines need attention is: [ boxed{0.003} ] 2. The probability that at least one machine needs attention is: [ boxed{0.997} ]

answer:1. We need to find the total number of minimally competent subsets of the set ( A = {1, 2, 3, ldots, 10} ). A subset is minimally competent if it contains ( n ) as an element and none of its proper subsets are competent. 2. For a subset to be minimally competent with respect to ( n ), it must contain ( n ) and all elements in the subset must be greater than ( n ). This ensures that no proper subset can be competent. 3. Let's consider each ( n ) from 1 to 10: - For ( n = 1 ), the minimally competent subset is ({1}). - For ( n = 2 ), the minimally competent subsets are of the form ({2, x}) where ( x > 2 ). The possible values for ( x ) are ( 3, 4, ldots, 10 ). There are ( 8 ) such subsets. - For ( n = 3 ), the minimally competent subsets are of the form ({3, x, y}) where ( x, y > 3 ). The number of such subsets is (binom{7}{1} = 7). - For ( n = 4 ), the minimally competent subsets are of the form ({4, x, y, z}) where ( x, y, z > 4 ). The number of such subsets is (binom{6}{2} = 15). - For ( n = 5 ), the minimally competent subsets are of the form ({5, x, y, z, w}) where ( x, y, z, w > 5 ). The number of such subsets is (binom{5}{3} = 10). - For ( n = 6 ), the minimally competent subsets are of the form ({6, x, y, z, w, v}) where ( x, y, z, w, v > 6 ). The number of such subsets is (binom{4}{4} = 1). - For ( n = 7 ), the minimally competent subsets are of the form ({7, x, y, z, w, v, u}) where ( x, y, z, w, v, u > 7 ). The number of such subsets is (binom{3}{2} = 3). - For ( n = 8 ), the minimally competent subsets are of the form ({8, x, y, z, w, v, u, t}) where ( x, y, z, w, v, u, t > 8 ). The number of such subsets is (binom{2}{1} = 2). - For ( n = 9 ), the minimally competent subsets are of the form ({9, x, y, z, w, v, u, t, s}) where ( x, y, z, w, v, u, t, s > 9 ). The number of such subsets is (binom{1}{0} = 1). - For ( n = 10 ), the minimally competent subset is ({10}). 4. Summing up all the minimally competent subsets, we get: [ binom{9}{0} + binom{8}{1} + binom{7}{2} + binom{6}{3} + binom{5}{4} + binom{4}{5} + binom{3}{6} + binom{2}{7} + binom{1}{8} + binom{0}{9} = 1 + 8 + 21 + 35 + 35 + 21 + 7 + 1 = 129 ] The final answer is (boxed{129})

answer:To solve n equiv -4702 pmod{8}, we first find the equivalence of -4702 modulo 8. We calculate: [ -4702 div 8 = -587.75 ] The integer part is -588, and multiplying back, we get: [ -588 times 8 = -4704 ] [ -4702 - (-4704) = 2 ] Thus, -4702 equiv 2 pmod{8}. Therefore, the integer n that satisfies 0 le n le 7 and n equiv -4702 pmod{8} is: [ n = boxed{2} ]

answer:The original expression can be rewritten as 2sin 2x cdot sin x + cos(2x + x). This simplifies to 2sin 2x cdot sin x + cos 2x cdot cos x - sin 2x cdot sin x. Further simplification gives cos 2x cdot cos x + sin 2x cdot sin x. This can be rewritten as cos(2x - x) = cos x. Therefore, the answer is boxed{cos x}.