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question:Coach Quick runs a basketball team called the Dashers, which has a roster of 14 players. Among them, Cyril, Pete, and Elaine refuse to play together in any combination. How many different starting lineups of 5 players can Coach Quick form if none of Cyril, Pete, and Elaine play together in a lineup?

answer:There are several cases for the starting lineup depending on which of Cyril, Pete, or Elaine is playing. Case 1: Cyril starts, but neither Pete nor Elaine start. Coach Quick needs to pick 4 players from the remaining 11 players (excluding Cyril, Pete, and Elaine). Thus, there are ( binom{11}{4} ) lineups. Case 2: Pete starts, but neither Cyril nor Elaine start. As in Case 1, 4 players are picked from the remaining 11 players. Hence, there are ( binom{11}{4} ) lineups. Case 3: Elaine starts, but neither Cyril nor Pete start. Again, 4 players come from the remaining 11 players. So, ( binom{11}{4} ) lineups are possible. Case 4: Neither Cyril, Pete, nor Elaine starts. There are 11 players available, from which all 5 players are to be picked. Therefore, ( binom{11}{5} ) lineups are possible. Summing these cases: binom{11}{4} + binom{11}{4} + binom{11}{4} + binom{11}{5} = 330 + 330 + 330 + 462 = 1452. Conclusion: The total number of different lineups Coach Quick can form without Cyril, Pete, and Elaine playing together is (boxed{1452}).

question:Given condition p: x^2 1 > 0, and condition q: x < 2, then neg p is to neg q as ( ) A: Sufficient but not necessary condition B: Necessary but not sufficient condition C: Both sufficient and necessary condition D: Neither sufficient nor necessary conditio

answer:The solution set for p is x > 1 or x < -1, so the solution set for neg p is -1 leq x leq 1, Condition q: x < -2, so neg q: x geq -2, Since the set of neg p is a subset of neg q, neg p is a sufficient but not necessary condition for neg q. Therefore, the correct choice is boxed{A}.

question:April has five different basil plants and four different tomato plants. In how many ways can she arrange the plants in a row if she puts all the tomato plants next to each other?

answer:1. Consider the five basil plants and the group of four tomato plants as one item. This gives us 6 items to arrange (5 basil + 1 tomato group). 2. Arrange these 6 items in a row, which can be done in 6! = 720 ways. 3. Then, arrange the four tomato plants within their group. This can be done in 4! = 24 ways. 4. Multiply these together to get the total arrangements: 720 times 24 = 17280. Conclusion with boxed answer: The number of ways April can arrange her plants with all tomato plants next to each other is boxed{17280}.

question:Evaluate the expression 2x^2 3x + 4 when x = 2.

answer:Given the expression 2x^2 - 3x + 4, we need to substitute x = 2 into the expression and simplify: [ 2x^2 - 3x + 4 = 2(2)^2 - 3(2) + 4. ] Calculating each term: - 2(2)^2 = 2 cdot 4 = 8, - -3(2) = -6, - Therefore, the expression simplifies to: [ 8 - 6 + 4 = 6. ] So, the value of the expression when x = 2 is boxed{6}.