answer：First, I need to determine the total number of spaces between the 16 trees. Since there are 16 trees, there will be 15 spaces between them.Next, I'll divide the total length of the sidewalk by the number of spaces to find the distance between each tree. So, 166 feet divided by 15 spaces equals approximately 11.07 feet between each tree.

answer：First, I need to solve the inequality (frac{y}{4} + frac{3}{7} < frac{2}{3}) to find the largest integer (y) that satisfies it.To eliminate the fractions, I'll multiply both sides of the inequality by the least common multiple of the denominators, which is 84:[84 left( frac{y}{4} + frac{3}{7} right) < 84 cdot frac{2}{3}]This simplifies to:[21y + 36 < 56]Next, I'll isolate (y) by subtracting 36 from both sides:[21y < 20]Finally, I'll divide both sides by 21 to solve for (y):[y < frac{20}{21}]Since (frac{20}{21}) is less than 1, the largest integer (y) that satisfies the inequality is 0.

answer：First, let's denote the number of toys Dhoni initially bought as ( x ).The average cost of these toys is 10, so the total cost of the toys Dhoni bought is:[text{Total Cost} = 10x]David buys an additional toy for 16, making the total number of toys 6. The average cost of these 6 toys is 11, so the total cost of all 6 toys is:[text{Total Cost} = 11 times 6 = 66]Now, we can set up the equation:[10x + 16 = 66]Solving for ( x ):[10x = 66 - 16][10x = 50][x = frac{50}{10} = 5]Therefore, Dhoni initially bought 5 toys.

answer：Okay, so Mary has a bag with 8 blue marbles and 7 red marbles. That means there are a total of 15 marbles in the bag. She's going to pick a marble, note its color, and put it back in the bag. She does this 7 times. I need to find the probability that exactly three of those picks are blue marbles.Hmm, since she's putting the marble back each time, each pick is independent. That means the outcome of one pick doesn't affect the others. So, this sounds like a binomial probability problem. In binomial problems, we have a fixed number of independent trials, each with two possible outcomes, and the probability of success is the same each time.In this case, each pick is a trial. The two outcomes are picking a blue marble (which I'll consider a "success") or picking a red marble (a "failure"). The probability of success (picking blue) is 8 out of 15, since there are 8 blue marbles. The probability of failure (picking red) is 7 out of 15.We want the probability of exactly 3 successes (blue marbles) in 7 trials. The formula for binomial probability is:P(k successes) = C(n, k) * p^k * (1-p)^(n-k)Where:- C(n, k) is the combination of n things taken k at a time.- p is the probability of success.- n is the number of trials.- k is the number of successes.So, plugging in the numbers:- n = 7- k = 3- p = 8/15- 1 - p = 7/15First, I need to calculate C(7, 3). That's the number of ways to choose 3 successes out of 7 trials. The formula for combinations is C(n, k) = n! / (k!(n - k)!).Calculating C(7, 3):7! / (3! * (7 - 3)!) = 7! / (3! * 4!) = (7 * 6 * 5 * 4!) / (3 * 2 * 1 * 4!) = (7 * 6 * 5) / (3 * 2 * 1) = 210 / 6 = 35.Okay, so C(7, 3) is 35.Next, I need to calculate p^k, which is (8/15)^3.Calculating (8/15)^3:8^3 = 51215^3 = 3375So, (8/15)^3 = 512/3375.Then, (1 - p)^(n - k) is (7/15)^(7 - 3) = (7/15)^4.Calculating (7/15)^4:7^4 = 240115^4 = 50625So, (7/15)^4 = 2401/50625.Now, putting it all together:P(3 blue marbles) = C(7, 3) * (8/15)^3 * (7/15)^4 = 35 * (512/3375) * (2401/50625).Let me compute this step by step.First, multiply 35 by 512/3375:35 * 512 = 18, 35 * 500 = 17,500 and 35 * 12 = 420, so total is 17,500 + 420 = 17,920.Wait, actually, 35 * 512: 35 * 500 = 17,500; 35 * 12 = 420; so 17,500 + 420 = 17,920.So, 35 * (512/3375) = 17,920 / 3,375.Now, simplify 17,920 / 3,375. Let's see if we can divide numerator and denominator by 5:17,920 ÷ 5 = 3,5843,375 ÷ 5 = 675So, 3,584 / 675.Can we simplify further? Let's check if 3,584 and 675 have any common factors.675 = 25 * 27 = 5^2 * 3^3.3,584: Let's see, 3,584 ÷ 2 = 1,792; ÷2 again = 896; ÷2 = 448; ÷2 = 224; ÷2 = 112; ÷2 = 56; ÷2 = 28; ÷2 = 14; ÷2 = 7. So, 3,584 = 2^9 * 7.675 is 5^2 * 3^3. No common factors with 3,584, so 3,584 / 675 is in simplest terms.Now, multiply this by 2,401 / 50,625.So, (3,584 / 675) * (2,401 / 50,625).Multiply numerators: 3,584 * 2,401.Hmm, that's a big number. Let me compute that.First, 3,584 * 2,000 = 7,168,000Then, 3,584 * 400 = 1,433,600Then, 3,584 * 1 = 3,584So, total is 7,168,000 + 1,433,600 = 8,601,600 + 3,584 = 8,605,184.Now, the denominator is 675 * 50,625.Compute 675 * 50,625.First, 675 * 50,000 = 33,750,000Then, 675 * 625 = ?Compute 675 * 600 = 405,000675 * 25 = 16,875So, 405,000 + 16,875 = 421,875So, total denominator is 33,750,000 + 421,875 = 34,171,875.So, the probability is 8,605,184 / 34,171,875.Now, let's see if we can simplify this fraction.First, check if both numerator and denominator are divisible by 3.8,605,184: Sum of digits: 8+6+0+5+1+8+4 = 32. 32 is not divisible by 3, so numerator isn't divisible by 3.Denominator: 3+4+1+7+1+8+7+5 = 36. 36 is divisible by 3, so denominator is divisible by 3, but numerator isn't, so we can't simplify further by 3.Check divisibility by 2: numerator is even, denominator is odd, so we can't divide by 2.Check divisibility by 5: numerator ends with 4, so no; denominator ends with 5, so yes, but numerator isn't, so no.Check divisibility by 7: Let's see.For numerator: 8,605,184 ÷ 7. Let's compute:7 * 1,229,312 = 8,605,184. So, numerator is divisible by 7.Denominator: 34,171,875 ÷ 7. Let's see:7 * 4,881,696 = 34,171,872, which is 3 less than 34,171,875. So, not divisible by 7.So, numerator is divisible by 7, denominator isn't. So, let's divide numerator by 7:8,605,184 ÷ 7 = 1,229,312.So, the fraction becomes 1,229,312 / 34,171,875.Check if this can be simplified further.Check divisibility by 3:1,229,312: 1+2+2+9+3+1+2 = 20. Not divisible by 3.34,171,875: 3+4+1+7+1+8+7+5 = 36. Divisible by 3, but numerator isn't.Check divisibility by 2: numerator is even, denominator is odd.Check divisibility by 5: numerator ends with 2, no; denominator ends with 5, yes.So, no further simplification possible.So, the probability is 1,229,312 / 34,171,875.Alternatively, we can write this as a decimal by dividing numerator by denominator.Let me compute 1,229,312 ÷ 34,171,875.First, note that 34,171,875 * 0.036 = ?Well, 34,171,875 * 0.03 = 1,025,156.2534,171,875 * 0.006 = 205,031.25So, 0.036 * 34,171,875 = 1,025,156.25 + 205,031.25 = 1,230,187.5But our numerator is 1,229,312, which is slightly less.So, 0.036 * 34,171,875 = 1,230,187.5Difference: 1,230,187.5 - 1,229,312 = 865.5So, 865.5 / 34,171,875 = approximately 0.0000253So, subtract that from 0.036: 0.036 - 0.0000253 ≈ 0.0359747So, approximately 0.035975, or about 3.5975%.So, the probability is approximately 3.6%.But since the question didn't specify the form, maybe we can leave it as a fraction.Alternatively, we can write it in terms of powers:But I think the exact fraction is 35 * (8/15)^3 * (7/15)^4 = 35 * 512/3375 * 2401/50625 = 35 * 512 * 2401 / (3375 * 50625)Which is 35 * 512 * 2401 / (3375 * 50625)But 35 is 5*7, 512 is 2^9, 2401 is 7^4, 3375 is 15^3 = (3*5)^3 = 3^3 *5^3, and 50625 is 15^4 = 3^4 *5^4.So, writing all in prime factors:Numerator: 5 *7 * 2^9 *7^4 = 2^9 *5 *7^5Denominator: 3^3 *5^3 *3^4 *5^4 = 3^(3+4) *5^(3+4) = 3^7 *5^7So, the fraction is (2^9 *5 *7^5) / (3^7 *5^7) = (2^9 *7^5) / (3^7 *5^6)But that might not be necessary unless asked.So, the exact probability is 35 * (8/15)^3 * (7/15)^4, which simplifies to 1,229,312 / 34,171,875, approximately 3.6%.

answer：First, I need to multiply the numerators of the fractions together and the denominators together.So, I'll calculate 1 times 1 times 3 times 5 for the numerators, which gives me 15.For the denominators, I'll multiply 3 times 2 times 4 times 6, resulting in 144.Now, I have the fraction 15 over 144.To simplify this fraction, I'll find the greatest common divisor (GCD) of 15 and 144, which is 3.Finally, I'll divide both the numerator and the denominator by their GCD, 3, to get the simplified fraction 5 over 48.

answer：First, I need to understand what the original event "hitting the target at least once" entails. This means that in two shots, the person either hits the target once or twice.To find the opposite event, I consider the scenario where the original event does not occur. In this case, the opposite would be missing the target entirely in both shots.Next, I evaluate the given options:- **A: At most one hit** – This includes scenarios where there is one hit or no hits, which overlaps with the original event.- **B: Both shots hit** – This is a subset of the original event and not the opposite.- **C: Both shots miss** – This directly represents the scenario where the original event does not occur.- **D: Only one shot hits** – This is part of the original event and not the opposite.Therefore, the correct opposite event is "Both shots miss."