💡First, identify the number of cut-outs Melody planned to paste on each card, which is 4.Next, determine the total number of cards she made, which is 6.To find the total number of cut-outs, multiply the number of cut-outs per card by the number of cards: 4 × 6 = 24.Therefore, Melody made 24 cut-outs of Christmas trees.

💡First, calculate the total number of CDs that store A can hold:Store A:- Number of shelves = 5- Number of CD racks per shelf = 7- Number of CDs per rack = 8Total CDs in store A = 5 * 7 * 8 = 280Next, calculate the total number of CDs that store B can hold:Store B:- Number of shelves = 4- Number of CD racks per shelf = 6- Number of CDs per rack = 7Total CDs in store B = 4 * 6 * 7 = 168Finally, add the total number of CDs from both stores together:Total CDs in store A and B = 280 + 168 = 448

💡Okay, so I have this probability problem about a beverage company testing an employee. They have 5 cups of beverages—3 are Beverage A and 2 are Beverage B. The employee has to taste them and select the 3 cups of Beverage A. Depending on how many they get right, they get evaluated as "Excellent," "Good," or "Qualified." The person can't tell the difference between A and B, so they're just guessing.First, I need to figure out the probability of getting "Excellent," which means selecting all 3 correct cups of Beverage A. Then, I also need to find the probability of being evaluated as "Good" or better, which includes both "Excellent" and "Good."Alright, let's start with part (1): the probability of being "Excellent." So, the employee needs to pick all 3 correct cups out of 5. Since they can't tell the difference, it's purely a matter of chance.I remember that when dealing with probabilities like this, especially when selecting items without replacement, combinations are useful. The total number of ways to choose 3 cups out of 5 is given by the combination formula:[C(n, k) = frac{n!}{k!(n - k)!}]Where ( n ) is the total number of items, and ( k ) is the number of items to choose.So, the total number of possible ways to select 3 cups from 5 is:[C(5, 3) = frac{5!}{3!(5 - 3)!} = frac{5 times 4 times 3!}{3! times 2!} = frac{5 times 4}{2 times 1} = 10]So, there are 10 possible ways to choose any 3 cups.Now, how many of these ways result in selecting all 3 correct Beverage A cups? Well, there's only one specific combination where all 3 selected cups are Beverage A. That's because Beverage A is only in cups 1, 2, and 3 (assuming they're labeled that way). So, the only correct combination is selecting cups 1, 2, and 3.Therefore, the number of successful outcomes is 1.So, the probability of being "Excellent" is the number of successful outcomes divided by the total number of possible outcomes:[P(text{Excellent}) = frac{1}{10}]Okay, that seems straightforward. Now, moving on to part (2): the probability of being evaluated as "Good" or better. "Good" means selecting exactly 2 correct cups of Beverage A. So, we need to calculate the probability of selecting exactly 2 correct cups and then add it to the probability of being "Excellent" to get the total probability of being "Good" or better.First, let's find the number of ways to select exactly 2 correct cups of Beverage A and 1 incorrect cup (which would be Beverage B).To do this, we can think of it as two separate choices:1. Choosing 2 correct cups from the 3 available Beverage A cups.2. Choosing 1 incorrect cup from the 2 available Beverage B cups.The number of ways to choose 2 correct cups from 3 is:[C(3, 2) = frac{3!}{2!(3 - 2)!} = frac{3 times 2!}{2! times 1!} = 3]So, there are 3 ways to choose 2 correct Beverage A cups.Next, the number of ways to choose 1 incorrect cup from 2 Beverage B cups is:[C(2, 1) = frac{2!}{1!(2 - 1)!} = frac{2 times 1!}{1! times 1!} = 2]So, there are 2 ways to choose 1 incorrect Beverage B cup.Since these two choices are independent, we multiply the number of ways to get the total number of successful outcomes for selecting exactly 2 correct cups:[3 times 2 = 6]Therefore, there are 6 ways to select exactly 2 correct cups of Beverage A and 1 incorrect cup.Now, to find the probability of being "Good," we take the number of successful outcomes (6) and divide by the total number of possible outcomes (10):[P(text{Good}) = frac{6}{10} = frac{3}{5}]But wait, the question asks for the probability of being evaluated as "Good" or better. "Better" in this context would include "Excellent." So, we need to add the probability of "Excellent" to the probability of "Good."We already calculated ( P(text{Excellent}) = frac{1}{10} ) and ( P(text{Good}) = frac{6}{10} ).So, adding these together:[P(text{Good or better}) = P(text{Excellent}) + P(text{Good}) = frac{1}{10} + frac{6}{10} = frac{7}{10}]Therefore, the probability of being evaluated as "Good" or better is ( frac{7}{10} ).Let me just double-check my calculations to make sure I didn't make any mistakes.For part (1), total combinations are 10, and only 1 is all correct. That seems right.For part (2), choosing 2 correct and 1 incorrect: 3 ways to choose 2 correct, 2 ways to choose 1 incorrect, so 6 ways total. That makes sense. Then, adding the 1 way for "Excellent," we get 7 successful outcomes out of 10. Yes, that seems correct.I think I've covered all the necessary steps and didn't skip over anything important. The logic flows correctly, and the calculations seem accurate. So, I feel confident about these results.

💡First, determine the number of shirts Razel received by multiplying the number of shirts Hazel received by 2.Next, add the number of shirts Hazel and Razel received to find the total number of shirts they have together.

💡Okay, so I have this geometry problem here about two planes, α and β, and a line a that's contained within plane α. The question is asking about the condition where line a is perpendicular to plane β, and how that relates to the condition that plane α is perpendicular to plane β. The options are whether this is a sufficient condition, a necessary condition, both, or neither.Alright, let me start by recalling some basic concepts. When we talk about two planes being perpendicular, it means that the angle between them is 90 degrees. In three-dimensional space, two planes are perpendicular if their normal vectors are perpendicular to each other. That is, if the dot product of their normal vectors is zero.Now, the line a is in plane α. If line a is perpendicular to plane β, what does that imply? Well, if a line is perpendicular to a plane, it means that it's perpendicular to every line in that plane. So, if line a is perpendicular to plane β, then it must be perpendicular to every line in plane β, including any lines that might be in both plane β and plane α.Wait, but plane α and plane β could intersect along a line. If line a is in plane α and is perpendicular to plane β, then it must be perpendicular to the line of intersection of α and β. That seems important.I think there's a theorem related to this. If a line in one plane is perpendicular to another plane, then the two planes are perpendicular. So, if line a is in plane α and is perpendicular to plane β, then plane α must be perpendicular to plane β. That makes sense because the line a would be acting as a sort of "indicator" that the planes are at a right angle to each other.But is the converse true? If plane α is perpendicular to plane β, does that mean that any line in plane α is perpendicular to plane β? Hmm, no, I don't think so. If two planes are perpendicular, it means that there exists at least one line in each plane that is perpendicular to the other plane. But not every line in plane α will necessarily be perpendicular to plane β. Only those lines that are perpendicular to the line of intersection of the two planes will be perpendicular to the other plane.So, in this case, if line a is perpendicular to plane β, then plane α is definitely perpendicular to plane β. But plane α being perpendicular to plane β doesn't necessarily mean that line a is perpendicular to plane β. Line a could be any line in plane α, which might not be perpendicular to plane β.Therefore, the condition that line a is perpendicular to plane β is sufficient to conclude that plane α is perpendicular to plane β. However, it's not necessary because plane α could be perpendicular to plane β without line a being perpendicular to plane β.So, putting it all together, the condition is sufficient but not necessary. That would correspond to option A.Wait, hold on. Let me double-check. If line a is perpendicular to plane β, then plane α must be perpendicular to plane β. So, it's a sufficient condition. But plane α being perpendicular to plane β doesn't require line a to be perpendicular; it just requires that there exists some line in α that's perpendicular to β. So, line a could be any line, not necessarily the one that's perpendicular.Therefore, yes, it's a sufficient condition but not a necessary one. So, the answer should be A.But wait, I'm a bit confused because sometimes these things can be tricky. Let me think of an example. Suppose plane α is the xy-plane and plane β is the xz-plane. They are perpendicular. Now, take line a in the xy-plane. If line a is along the x-axis, it's not perpendicular to plane β (xz-plane). But if line a is along the y-axis, then it is perpendicular to plane β. So, in this case, line a being perpendicular to plane β (if it's the y-axis) is sufficient to make plane α perpendicular to plane β. But plane α is already perpendicular to plane β regardless of whether line a is the y-axis or not. So, line a being perpendicular is not necessary, just sufficient.Yeah, that makes sense. So, I think I was right the first time. The answer is A.**Final Answer**boxed{A}

💡Okay, so I have this problem here: I need to find the value of (2 - i) divided by (1 + i), where i is the imaginary unit. Hmm, imaginary numbers can be a bit tricky, but I think I remember some rules about them. Let me try to work through this step by step.First, I know that when you have a complex number in the denominator, like (1 + i), it's often helpful to get rid of the imaginary part in the denominator. I think the way to do that is by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of (1 + i) should be (1 - i), right? Because the conjugate changes the sign of the imaginary part.So, if I multiply both the top and the bottom by (1 - i), that should help simplify things. Let me write that down:(2 - i)/(1 + i) * (1 - i)/(1 - i)Okay, so now I have to multiply out the numerators and the denominators. Let's start with the numerator: (2 - i) times (1 - i). I'll use the distributive property, which is also known as the FOIL method for binomials.First, I'll multiply 2 by 1, which is 2. Then, 2 times -i is -2i. Next, -i times 1 is -i, and finally, -i times -i is... hmm, that's i squared. Wait, i squared is equal to -1, right? So, putting it all together:2 * 1 = 22 * (-i) = -2i(-i) * 1 = -i(-i) * (-i) = i² = -1So, adding all those up: 2 - 2i - i - 1. Let me combine like terms. The real parts are 2 and -1, which gives me 1. The imaginary parts are -2i and -i, which adds up to -3i. So, the numerator simplifies to 1 - 3i.Now, let's work on the denominator: (1 + i) times (1 - i). Again, I'll use the distributive property.1 * 1 = 11 * (-i) = -ii * 1 = ii * (-i) = -i²So, adding those up: 1 - i + i - i². Hmm, the -i and +i cancel each other out, leaving me with 1 - i². Since i² is -1, this becomes 1 - (-1), which is 1 + 1, so 2.So, putting it all together, the numerator is 1 - 3i and the denominator is 2. Therefore, the fraction simplifies to (1 - 3i)/2.I can write this as two separate fractions: 1/2 - (3i)/2. So, the final answer should be 1/2 minus three halves of i.Wait, let me double-check my steps to make sure I didn't make a mistake. Starting with multiplying numerator and denominator by the conjugate, that seems right. Then expanding both, I think I did that correctly. In the numerator, I had 2 - 2i - i -1, which simplifies to 1 - 3i. In the denominator, I had 1 - i², which is 2. Yeah, that seems correct.I guess another way to check is to plug the original expression into a calculator that handles complex numbers, but since I don't have one handy, I'll just go over it again.Original expression: (2 - i)/(1 + i)Multiply numerator and denominator by (1 - i):Numerator: (2 - i)(1 - i) = 2*1 + 2*(-i) + (-i)*1 + (-i)*(-i) = 2 - 2i - i + i² = 2 - 3i -1 = 1 - 3iDenominator: (1 + i)(1 - i) = 1*1 + 1*(-i) + i*1 + i*(-i) = 1 - i + i - i² = 1 - i² = 1 - (-1) = 2So, yes, it's 1 - 3i over 2, which is 1/2 - (3/2)i. That seems consistent.I think I'm confident with this answer now. It's always good to go through the steps again to make sure I didn't skip anything or make a calculation error. Especially with imaginary numbers, it's easy to mix up the signs or forget that i squared is -1.So, in summary, by multiplying both the numerator and denominator by the conjugate of the denominator, I was able to simplify the expression and eliminate the imaginary unit from the denominator, resulting in the simplified form of 1/2 - (3/2)i.