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I understand that the pigeonhole principle is supposedly a quite simple concept. However could you please explain to me the reasoning of how you reach this answer. Thank you.

Question: A basket cannot contain more than $24$ apples. What is the minimum amount of baskets you must have, to ensure you have at least $5$ baskets with the same number of apples in them (all baskets have at least $1$ apple contained within).

Answer of this question being $97$ baskets.

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The basic idea of the pigeonhole principle is trivial , but the application can be much more difficult.

Main idea : If we distribute $n+1$ pigeons among $n$ cages, at least one cage must have more than one pigeon.

The problem here :

There are $24$ possibilities for the number of apples in a basket.

Therefore, $96$ baskets cannot be enough because every number from $1$ to $24$ can appear exactly four times.

But if we add another basket, it is not possible anymore that all the numbers appear at most $4$ times because then, at most $96$ baskets would be possible.

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Why $96$ is not enought:

For every number $x$ between $1$ and $24$ you take $4$ baskets with $x$ apples inside. Then you have $4\cdot 24 = 96$ baskets. By construction, there is no number such that $5$ baskets have this number of apples.

On the other hand if you have $97$ baskets and assume there are maximum $4$ baskets with the same number of apples. Then again the number of baskets is limited to $4\cdot 24$ which is less than your number of baskets. So you have a contradiction. Hence, there is no configuration with max $4$ baskets with the same number of apples for $97$ baskets.

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Since a basket cannot contain more than $24$ apples, we consider a basket to be a collection of $24$ pigeonholes. Hence if we have $4$ baskets, there are altogether $24 \times 4 = 96$ pigeonholes. Indeed $96$ pigeonholes can contain $96$ pigeons (apples). But if we have $97$ pigeons, by pigeonhole principle, at least one of the pigeonhole must have two pigeons, which is not allowed (the $4$ baskets are full, so we need an extra one). Hence $97$ is the answer.

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The number of apples in a basket is between 1 and 24. That's 24 different values.

Suppose that in our collection of baskets, none of these numbers occurs at least five times.

This means that in our collection of baskets each of these numbers occurs at most four times.

But then we can have at most 24 times 4 baskets.

Now, $24 \times 4 = 96$. So if we have more than that number of baskets, that is if we have at least 97 baskets, then one of those numbers between 1 and 24 must occur at least five times, that is then we have at least five baskets with the same number of apples.

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There are different versions of PigeonHole Principle, but they all basically amount to:

if number of containers is exceeded by number of objects, then some container has something happen with certainty (or certain probability in a non-equiprobable case)

The logic of this statement, can be done via repeated use of the pigeonhole principle with strict limits. If We have a container that can hold 24 baskets( last container can be partially filled), each one having a different number of objects, 1 through 24, it will take 5 of these large containers to have 5 baskets with the same amount at most. 4 large containers are not enough but 4 containers contain $4\cdot 24 =96$ baskets. only adding 1 more basket in a fifth container, forces a collection of 5. Because, each of the first 4 containers had a basket with each allowed amount in it. adding that basket gets us 97 baskets put into 5 containers.

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