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Find the number of ways that Magnus can give out 12 identical stickers to 8 of his friends. (Not everyone has to get a sticker.)

Find the number of ways that Magnus can give out 12 identical stickers to 8 of his friends, if every friend gets at least one sticker.

Apr 17, 2024

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There seems to be an error in the provided code. The result shows 0, which is incorrect.

Let's solve this problem using a dynamic programming approach.

We can define a 2D table dp where dp[i][j] represents the number of ways to distribute i stickers among j friends.

Base Case: If there are no stickers (i = 0), there's only one way (no distribution) for any number of friends (j). So, initialize dp[0][j] to 1 for all j.

Inductive Case: For a given number of stickers (i) and friends (j), we have two options:

Give the current friend a sticker (option 1). In this case, we use the number of ways to distribute the remaining stickers (i - 1) among the remaining friends (j - 1). This is captured by dp[i - 1][j - 1].

Don't give the current friend a sticker (option 2). In this case, we use the number of ways to distribute the same number of stickers (i) among the same number of friends (j). This is captured by dp[i][j - 1].

The total number of ways for dp[i][j] is the sum of these two options.

Implementation:

Python

def distribute_stickers(stickers, friends): """ This function calculates the number of ways to distribute a given number of identical stickers among a specified number of friends, where some friends may not receive any stickers. Args: stickers (int): The total number of identical stickers. friends (int): The number of friends. Returns: int: The number of ways to distribute the stickers. """ # Use dynamic programming to solve the problem. # dp[i][j] represents the number of ways to distribute i stickers among j friends. dp = [[0 for _ in range(friends + 1)] for _ in range(stickers + 1)] # Base case: If there are no stickers, there's only one way (no distribution). for friend in range(friends + 1): dp[0][friend] = 1 # Iterate through the number of stickers and friends. for sticker in range(1, stickers + 1): for friend in range(1, friends + 1): # Option 1: Give the current friend a sticker. dp[sticker][friend] += dp[sticker - 1][friend - 1] # Option 2: Don't give the current friend a sticker. dp[sticker][friend] += dp[sticker][friend - 1] # The final answer is the number of ways to distribute all stickers (stickers) among all friends (friends). return dp[stickers][friends] # Example usage: Find the number of ways to distribute 12 stickers among 8 friends. number_of_ways = distribute_stickers(12, 8) print(number_of_ways)

Use code with caution.

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Explanation:

The code iterates through the number of stickers and friends, filling the dp table. For each cell dp[i][j], it calculates the sum of the two options mentioned above and stores it in the cell. Finally, dp[stickers][friends] gives us the total number of ways to distribute 12 stickers among 8 friends.

This approach correctly considers the scenario where some friends might not receive stickers.

Apr 17, 2024