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14 May 2024, 22:13
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How to solve the following questions?
Bunuel kindly move or delete the post as you deem fit.
- In how many ways can 5 different candies be distributed in 4 similar baskets?
- In how many ways can 5 similar candies be distributed in 4 similar baskets?
- In how many ways can 5 different candies be distributed in 4 idifferent baskets?
- In how many ways can 5 similar candies be distributed in 4 different baskets
Bunuel kindly move or delete the post as you deem fit.
21 May 2024, 00:00
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Sure, Noida. Here’s how to solve your questions:
Since the baskets are similar (identical), we only care about the number of candies in each basket, not which specific basket they are in. This problem reduces to finding the partitions of 5 candies among 4 baskets.
We can enumerate the possible distributions manually since 5 is a small number:
All 5 candies in one basket: 1 way ([5])
4 candies in one basket, 1 in another: 1 way ([4, 1])
3 candies in one basket, 2 in another: 1 way ([3, 2])
3 candies in one basket, 1 in two others: 1 way ([3, 1, 1])
2 candies in two baskets, 1 in another: 1 way ([2, 2, 1])
2 candies in one basket, 1 in three others: 1 way ([2, 1, 1, 1])
1 candy in each basket: 1 way ([1, 1, 1, 1, 1])
Therefore, there are 7 ways to distribute 5 different candies into 4 similar baskets.
This is equivalent to finding the number of partitions of 5 into at most 4 parts:
5: [5]
4 + 1: [4, 1]
3 + 2: [3, 2]
3 + 1 + 1: [3, 1, 1]
2 + 2 + 1: [2, 2, 1]
2 + 1 + 1 + 1: [2, 1, 1, 1]
1 + 1 + 1 + 1 + 1: [1, 1, 1, 1, 1]
Again, there are 7 ways.
Each candy can go into any of the 4 baskets, so we have 4 choices for each of the 5 candies. This can be calculated as:
4^5=1024
For 5 similar candies into 4 different baskets, there are 56 ways using the stars and bars method.
Since the baskets are similar (identical), we only care about the number of candies in each basket, not which specific basket they are in. This problem reduces to finding the partitions of 5 candies among 4 baskets.
We can enumerate the possible distributions manually since 5 is a small number:
All 5 candies in one basket: 1 way ([5])
4 candies in one basket, 1 in another: 1 way ([4, 1])
3 candies in one basket, 2 in another: 1 way ([3, 2])
3 candies in one basket, 1 in two others: 1 way ([3, 1, 1])
2 candies in two baskets, 1 in another: 1 way ([2, 2, 1])
2 candies in one basket, 1 in three others: 1 way ([2, 1, 1, 1])
1 candy in each basket: 1 way ([1, 1, 1, 1, 1])
Therefore, there are 7 ways to distribute 5 different candies into 4 similar baskets.
This is equivalent to finding the number of partitions of 5 into at most 4 parts:
5: [5]
4 + 1: [4, 1]
3 + 2: [3, 2]
3 + 1 + 1: [3, 1, 1]
2 + 2 + 1: [2, 2, 1]
2 + 1 + 1 + 1: [2, 1, 1, 1]
1 + 1 + 1 + 1 + 1: [1, 1, 1, 1, 1]
Again, there are 7 ways.
Each candy can go into any of the 4 baskets, so we have 4 choices for each of the 5 candies. This can be calculated as:
4^5=1024
For 5 similar candies into 4 different baskets, there are 56 ways using the stars and bars method.
