💡First, I need to determine how long it takes for Pipe A to fill the tank. Pipe A can fill the tank in 32 minutes.Next, since Pipe B is 3 times faster than Pipe A, I can calculate Pipe B's filling time by dividing Pipe A's time by 3. So, Pipe B takes 32 minutes divided by 3, which is approximately 10.67 minutes.Now, I'll find the filling rates of both pipes. Pipe A's rate is 1 tank per 32 minutes, which is 1/32 tank per minute. Pipe B's rate is 1 tank per 10.67 minutes, which is approximately 0.0938 tank per minute.When both pipes are working together, their rates add up. Adding Pipe A's rate (1/32) and Pipe B's rate (3/32) gives a combined rate of 4/32, which simplifies to 1/8 tank per minute.Finally, to find out how long it takes to fill the tank together, I take the reciprocal of the combined rate. So, 1 divided by 1/8 equals 8 minutes. Therefore, it will take 8 minutes for both pipes to fill the tank together.

💡Okay, so I need to calculate the sum of the series 2 + 4 + 6 + ... + 18 + 20. Hmm, let me think about how to approach this. I remember that when you have a series where each term increases by the same amount, it's called an arithmetic series. In this case, each term increases by 2, so that fits.First, I should figure out how many terms are in this series. The first term is 2, and the last term is 20. Since each term increases by 2, I can find the number of terms by dividing the difference between the last and first term by the common difference and then adding 1. So, (20 - 2) / 2 = 18 / 2 = 9, and then adding 1 gives me 10 terms. Okay, so there are 10 terms in this series.Now, to find the sum of an arithmetic series, I recall there's a formula: S = n/2 * (a1 + an), where n is the number of terms, a1 is the first term, and an is the last term. Plugging in the values I have, n is 10, a1 is 2, and an is 20. So, S = 10/2 * (2 + 20) = 5 * 22 = 110. Wait, that seems straightforward. Let me double-check to make sure I didn't make a mistake.Alternatively, I can add the terms one by one to verify. Starting from 2, adding 4 gives me 6, then adding 6 gives 12, then adding 8 gives 20, adding 10 gives 30, adding 12 gives 42, adding 14 gives 56, adding 16 gives 72, adding 18 gives 90, and finally adding 20 gives 110. Okay, that matches what I got using the formula. So, I think 110 is the correct sum.Another way to think about it is that this series is the sum of the first 10 even numbers. I remember that the sum of the first n even numbers is n(n + 1). Let me test that formula with n = 10. So, 10 * (10 + 1) = 10 * 11 = 110. Yep, that's the same result. So, that gives me more confidence that 110 is the right answer.Wait, just to make sure I didn't confuse it with the formula for the sum of the first n natural numbers, which is n(n + 1)/2. But since we're dealing with even numbers, it's actually n(n + 1). Let me confirm that. For example, the sum of the first 2 even numbers is 2 + 4 = 6, and using the formula, 2 * (2 + 1) = 6. That works. Similarly, the first 3 even numbers: 2 + 4 + 6 = 12, and 3 * (3 + 1) = 12. Perfect, so the formula holds.Therefore, applying it to n = 10, I get 10 * 11 = 110. So, all methods point to the same answer, which is 110. I think I'm confident that's correct.Just to recap, the steps I took were:1. Recognize that the series is arithmetic with a common difference of 2.2. Determine the number of terms by calculating (last term - first term) / common difference + 1, which gave me 10 terms.3. Use the arithmetic series sum formula: S = n/2 * (a1 + an), resulting in 110.4. Verified by manually adding the terms step by step, arriving at the same result.5. Cross-checked using the formula for the sum of the first n even numbers, which also gave 110.So, after multiple methods and checks, I'm pretty sure the sum is 110.

💡First, I need to determine the number of choices for selecting a goalkeeper. Since one team member cannot play as a goalkeeper, there are 15 available players for this position.Next, after selecting the goalkeeper, there are 15 remaining players who can be chosen as defenders.Then, I need to choose a midfielder from the remaining 14 players.Finally, I will select a forward from the remaining 13 players.To find the total number of ways to choose the starting lineup, I multiply the number of choices for each position: 15 (goalkeepers) × 15 (defenders) × 14 (midfielders) × 13 (forwards).Calculating this gives 15 × 15 × 14 × 13 = 42,210.Therefore, the total number of ways to choose the starting lineup is 42,210.

💡Okay, so I'm trying to figure out how many different "new" ice cream flavors Ice-cream-o-rama can create by combining five scoops from their four basic flavors: chocolate, vanilla, strawberry, and mint. Hmm, this sounds like a combinatorics problem, specifically about combinations with repetition. Let me think through this step by step.First, I know that when we're dealing with combinations where repetition is allowed, the formula to use is the "stars and bars" theorem. This theorem helps us determine the number of ways to distribute identical items (in this case, scoops of ice cream) into distinct categories (the different flavors). The formula for combinations with repetition is:[C(n + k - 1, k)]Where:- ( n ) is the number of types of items (flavors, in this case) we're choosing from.- ( k ) is the number of items (scoops) we're choosing.In this problem, we have 4 flavors (( n = 4 )) and we want to create combinations using 5 scoops (( k = 5 )). So, plugging these values into the formula:[C(4 + 5 - 1, 5) = C(8, 5)]Now, I need to calculate ( C(8, 5) ), which is the number of combinations of 8 items taken 5 at a time. The combination formula is:[C(n, k) = frac{n!}{k!(n - k)!}]So, substituting the values:[C(8, 5) = frac{8!}{5!(8 - 5)!} = frac{8!}{5! times 3!}]Calculating the factorials:- ( 8! = 40320 )- ( 5! = 120 )- ( 3! = 6 )Plugging these back into the equation:[C(8, 5) = frac{40320}{120 times 6} = frac{40320}{720} = 56]So, there are 56 different ways to combine five scoops from four basic flavors. This means Ice-cream-o-rama can create 56 unique "new" flavors by varying the number of scoops of each basic flavor in their combinations.Wait, let me double-check to make sure I didn't make a mistake. Sometimes with combinatorics, it's easy to mix up the formula or miscount the values. Let me verify the steps again.1. **Identify the problem as combinations with repetition**: Yes, because the order of scoops doesn't matter, and we can have multiple scoops of the same flavor.2. **Apply the stars and bars formula**: Correct, ( C(n + k - 1, k) ).3. **Substitute the values**: ( n = 4 ), ( k = 5 ), so ( C(8, 5) ).4. **Calculate the combination**: ( C(8, 5) = 56 ).Everything seems to add up. I think 56 is the right answer. It's always good to think about a simpler case to verify. For example, if we had 2 flavors and 3 scoops, how many combinations would there be?Using the same formula:[C(2 + 3 - 1, 3) = C(4, 3) = 4]Listing them out:- 3 scoops of flavor 1- 2 scoops of flavor 1 and 1 of flavor 2- 1 scoop of flavor 1 and 2 of flavor 2- 3 scoops of flavor 2That's 4 combinations, which matches the formula. So, applying the same logic to the original problem seems valid.Another way to think about it is using the "stars and bars" visual. Imagine you have 5 stars (representing the scoops) and 3 bars (to separate the 4 flavors). The arrangement of stars and bars represents different distributions of scoops among flavors.For example:[***|*|*|]This could represent 3 scoops of chocolate, 1 of vanilla, 1 of strawberry, and 0 of mint.The total number of symbols is 5 stars + 3 bars = 8 symbols. The number of ways to arrange these is the number of ways to choose positions for the bars (or the stars), which is ( C(8, 3) ) or equivalently ( C(8, 5) ), both of which equal 56.This reinforces that the answer is indeed 56.I think I've covered all the bases here. The key was recognizing it as a combinations with repetition problem and correctly applying the stars and bars formula. It's also helpful to verify with simpler cases to ensure the logic holds.

💡Okay, so I've got this problem here about four friends: Wanda, Xander, Yusuf, and Zealand. Zealand didn't have any money at the start, but the others did. Then, Wanda gave Zealand one-sixth of her money, Xander gave one-fifth of his, and Yusuf gave one-fourth of his. The key point is that each of them gave Zealand the same amount. I need to figure out what fraction of the group's total money Zealand now has. The options are from (A) 1/8 to (E) 2/5.Alright, let me break this down. First, let's denote the amount each person gave to Zealand as some variable. Since they all gave the same amount, let's say each gave Zealand an amount of 'x'. So, Wanda gave x, Xander gave x, and Yusuf gave x. That means Zealand received a total of 3x.Now, I need to find out how much money each of Wanda, Xander, and Yusuf originally had. Starting with Wanda: she gave one-sixth of her money, which is x. So, if one-sixth of her money is x, then her total money must be 6x. Because (1/6) * 6x = x.Similarly, Xander gave one-fifth of his money, which is also x. So, his total money must be 5x. Because (1/5) * 5x = x.Yusuf gave one-fourth of his money, which is x. So, his total money must be 4x. Because (1/4) * 4x = x.So, to recap:- Wanda had 6x- Xander had 5x- Yusuf had 4x- Zealand had 0 initiallyNow, let's calculate the total money in the group before any transfers. That would be Wanda + Xander + Yusuf + Zealand. So, 6x + 5x + 4x + 0 = 15x.After the transfers, each of Wanda, Xander, and Yusuf has given away x. So, their new amounts are:- Wanda: 6x - x = 5x- Xander: 5x - x = 4x- Yusuf: 4x - x = 3x- Zealand: 0 + x + x + x = 3xSo, the total money in the group remains the same, which is 15x. Now, Zealand has 3x. So, the fraction of the total money that Zealand now has is 3x / 15x, which simplifies to 3/15, and that reduces to 1/5.Wait, let me double-check that. So, Wanda had 6x, gave x, so she has 5x left. Xander had 5x, gave x, so 4x left. Yusuf had 4x, gave x, so 3x left. Zealand had 0, received 3x, so 3x. Total is 5x + 4x + 3x + 3x = 15x. Yep, that's correct.So, Zealand's share is 3x out of 15x, which is indeed 1/5. So, the answer should be (C) 1/5.I think that makes sense. Each gave the same amount, and by figuring out their original amounts based on the fractions they gave, we can determine the total and Zealand's portion. It all checks out.

💡Okay, so I need to figure out what percent of his earnings John had left after spending on rent and a dishwasher. Let me start by understanding the problem step by step.First, John spent 40% of his earnings on rent. That seems straightforward. So if his total earnings are, say, 100, he spent 40 on rent. But I guess I should keep it in percentages for now to make it easier.Next, he spent 30% less than what he spent on rent to buy a new dishwasher. Hmm, okay, so he spent 30% less than the rent amount. Since he spent 40% on rent, 30% of that would be 0.3 * 40% = 12%. So, he spent 12% less than 40% on the dishwasher. Wait, no, that's not quite right. It says 30% less than what he spent on rent. So, if he spent 40% on rent, then 30% less than that would be 40% - (30% of 40%). Let me calculate that: 30% of 40% is 12%, so 40% - 12% = 28%. So, he spent 28% on the dishwasher.Now, to find out the total percentage he spent, I need to add the rent and the dishwasher expenses together. That would be 40% (rent) + 28% (dishwasher) = 68%. So, he spent a total of 68% of his earnings.To find out what percent he had left, I subtract the total spent from 100%. So, 100% - 68% = 32%. Therefore, John had 32% of his earnings left over.Wait, but I'm a bit confused. Did I calculate the dishwasher expense correctly? Let me double-check. He spent 30% less than what he spent on rent. So, if rent was 40%, then 30% of 40% is 12%, and subtracting that from 40% gives 28%. Yeah, that seems right. So, total spent is 68%, leaving 32% remaining.I think that's it. So, John had 32% of his earnings left after paying for rent and the dishwasher.