answer:To solve this, we first identify that three of the four digits must form an arithmetic sequence, and the fourth digit, which is distinct and non-zero, does not participate in the average relationship. 1. **Counting Arithmetic Sequences**: The digits range from 1 to 9 (since 0 is excluded). For a common difference of 1, the smallest digit can be 1 through 7 (to fit in three digits by 9), yielding 7 sequences. For a common difference of 2, 1 through 6 can be the first digit, giving 6 sequences, and so on. This counting goes until the common difference of 4, where only two sequences (1, 5, 9 and 2, 6, 8) are possible. 2. **Adding the Fourth Digit**: Each sequence of three digits can be augmented by adding one of the remaining six digits not used in the sequence. 3. **Permutations**: Each valid set of four digits can be arranged in (4!) (24) ways because all digits are distinct. - For common difference of 1: (7 times 6 times 24 = 1008) - For common difference of 2: (6 times 6 times 24 = 864) - For common difference of 3: (5 times 6 times 24 = 720) - For common difference of 4: (2 times 6 times 24 = 288) Total: (1008 + 864 + 720 + 288 = boxed{2880}) four-digit numbers.

answer:Start by calculating the inner square root: sqrt{25} = 5. Next, multiply the result of the inner square root by the other number in the product: 49 times 5 = 245. Finally, find the square root of the result: sqrt{245} = sqrt{49 times 5} = 7 times sqrt{5} = 7sqrt{5}. Since 7sqrt{5} is a simplified exact form, the final answer is: boxed{7sqrt{5}}.

answer:Given the equation ax-y=1 and the solutions x=1 and y=-3, we substitute these values into the equation to find the value of a. Starting with the substitution, we have: [a(1) - (-3) = 1] This simplifies to: [a + 3 = 1] To solve for a, we isolate a by subtracting 3 from both sides of the equation: [a = 1 - 3] Simplifying the right side gives us: [a = -2] Therefore, the value of a is boxed{-2}.

answer:1. **Square Calculation**: The vertices of the square are at the circle, meaning the square's diagonal is the diameter of the circle, 20 units. Using the property of squares with diagonal d, the side length s can be calculated as: [ s = frac{d}{sqrt{2}} = frac{20}{sqrt{2}} = 10sqrt{2} ] Consequently, the area A_{text{square}} of the square is: [ A_{text{square}} = s^2 = (10sqrt{2})^2 = 200 ] 2. **Circle Calculation**: The area A_{text{circle}} of the circle can be calculated using the formula pi r^2 where r is the radius: [ A_{text{circle}} = pi times 10^2 = 100pi ] 3. **Probability Calculation**: The probability P that the dart lands in the square is the ratio of the area of the square to that of the circle: [ P = frac{A_{text{square}}}{A_{text{circle}}} = frac{200}{100pi} = frac{2}{pi} ] Conclusion: The probability that the dart lands within the center square is frac{2{pi}}. The final answer is **B) boxed{frac{2}{pi}}**.