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To find the number of ways Magnus can give out 12 identical stickers to 2 of his friends, we can use a concept from combinatorics called "stars and bars." We represent the 12 stickers as 12 stars (*...), and we need to divide them among 2 friends by placing 1 divider between the stars for the first friend to receive the stickers and 1 divider between the stars for the second friend to receive the stickers.
 

For example, if Magnus gives all 12 stickers to the first friend, it would look like this: || (where * represents a sticker and | represents a divider). The number of ways to distribute the stickers can be calculated using the formula: n + k choose k, where n is the number of stars (stickers) and k is the number of dividers (friends). In this case, n = 12 stickers and k = 2 friends. Plugging these values into the formula: 14 choose 2 = 14! / (2! 12!) = 14 13 / (2 * 1) = 91

Therefore, Magnus can give out the 12 identical stickers to his 2 friends in 91 different ways.

1. Give one sticker to each friend, leaving 10 stickers to distribute.

2. Imagine the remaining 10 stickers as stars and 1 separator bar.

3. You have a total of 10 stars and 1 bar to divide the stickers.

4. Use the formula "n + k - 1 choose k - 1" where n is the number of stars and k is the number of bars.

5. Plug in n = 10 and k = 2 into the formula.

6. The number of ways to distribute the stickers among friends is 11.

Therefore, Magnus can give out the 12 identical stickers to his 2 friends in 11 different ways, ensuring each friend receives at least one sticker.

To calculate the probability that Dhruv gets exactly 7 chocolates out of the 8 chocolates Miyu is giving out to her 5 friends, we need to determine the total number of ways Dhruv can receive 7 chocolates and then divide that by the total number of ways the chocolates can be distributed.
 

1. Calculate the total number of ways Dhruv can receive 7 chocolates: - Dhruv can receive 7 chocolates in 1 way. - The remaining 1 chocolate can be given to any of the other 4 friends, which can happen in 4 ways.

2. Calculate the total number of ways the 8 chocolates can be distributed among the 5 friends: - Each chocolate can be given to any of the 5 friends, so there are 5^8 ways to distribute the chocolates.

3. Calculate the probability: - The probability that Dhruv gets exactly 7 chocolates is the number of ways Dhruv can get 7 chocolates divided by the total number of ways to distribute the chocolates. - Therefore, the probability is 4 / 5^8.

To find the number of solutions to the equation u + v + w + x + y + z = 2, where u, v, w, x, y, z are nonnegative integers and x is at most 1, we can break it down into cases based on the value of x:

1. If x = 0: In this case, the equation becomes u + v + w + y + z = 2. Since the sum of the nonnegative integers is 2, there are 6 choose 4 = 15 solutions to this equation.

2. If x = 1: In this case, the equation becomes u + v + w + y + z = 1. Similarly, there are 5 choose 4 = 5 solutions to this equation.
 

Therefore, the total number of solutions to the given equation is 15 + 5 = 20.

Given that 15 members can paint the wall in 45 minutes, we can calculate the rate at which one member paints the wall.Let's denote the rate at which one member paints the wall as R (wall per minute). If 15 members paint the wall in 45 minutes, then the combined rate of all 15 members is 15R. So, 15R = 1 wall painted in 45 minutes. Thus, the rate of one member painting the wall is R = 1/15 wall per minute.

Now, we need to find out how long it would take 20 members to paint the same wall. Let T be the time (in minutes) it would take 20 members to paint the wall. The combined rate of all 20 members is 20R, and we know that the rate of one member is R = 1/15 wall per minute.

Therefore, 20R = 20/15 = 4/3 walls per minute. We want to find out how long it would take 20 members to paint 1 wall, so we set up the equation: T * (4/3) = 1. Solving for T: T = 3/4 minutes.
 

Therefore, it would take 20 members 3/4 minutes to paint the wall.