What is the number of ways to distribute n distinct objects among r people such that the difference between the maximum and minimum number of objects a person can get is atmost 1?

Thanks in advance.

To solve the problem of distributing \( n \) distinct objects among \( r \) people such that the difference between the maximum and minimum number of objects any person receives is at most 1, we need to consider how the objects can be divided evenly among the people. This scenario typically arises when \( n \) is either divisible by \( r \) or leaves a remainder when divided by \( r \).

Understanding the Distribution

When distributing \( n \) distinct objects, the goal is to ensure that no person receives significantly more or fewer objects than another. This means that each person should receive either \( \lfloor \frac{n}{r} \rfloor \) or \( \lceil \frac{n}{r} \rceil \) objects. Let's break this down further:

Calculating the Distribution

• If \( n \) is divisible by \( r \) (i.e., \( n \mod r = 0 \)), then each person will receive exactly \( \frac{n}{r} \) objects.
• If \( n \) is not divisible by \( r \) (i.e., \( n \mod r \neq 0 \)), then some people will receive \( \lfloor \frac{n}{r} \rfloor \) objects, and the remaining people will receive \( \lceil \frac{n}{r} \rceil \) objects. Specifically, if \( n = qr + k \) (where \( q \) is the quotient and \( k \) is the remainder), then \( k \) people will receive \( q + 1 \) objects, and \( r - k \) people will receive \( q \) objects.

Counting the Ways

Now, let's calculate the number of ways to achieve this distribution:

Case 1: \( n \) is divisible by \( r \)

In this case, each person gets \( \frac{n}{r} \) objects. The number of ways to distribute \( n \) distinct objects to \( r \) people is given by:

Formula: \( r! \) (the number of ways to assign groups to people).

Case 2: \( n \) is not divisible by \( r \)

Let’s say \( n = qr + k \) where \( k \) is the remainder. Here’s how we can distribute the objects:

• Choose \( k \) people from \( r \) to receive \( q + 1 \) objects. This can be done in \( \binom{r}{k} \) ways.
• Now, we need to distribute the \( n \) distinct objects. The number of ways to assign \( q + 1 \) objects to each of the \( k \) people and \( q \) objects to the remaining \( r - k \) people can be calculated using the multinomial coefficient.

Formula: The total number of ways is given by:

Final Count: \( \binom{r}{k} \cdot \frac{n!}{(q + 1)!^k \cdot q!^{(r-k)}} \)

Example for Clarity

Let’s say we have 5 distinct objects and want to distribute them among 3 people. Here, \( n = 5 \) and \( r = 3 \). Since \( 5 \mod 3 = 2 \), we have:

• Each of the 2 people will receive \( 2 \) objects, and 1 person will receive \( 1 \) object.
• We can choose 2 people from 3 to receive 2 objects in \( \binom{3}{2} = 3 \) ways.
• Now, we distribute the objects: \( \frac{5!}{2!^2 \cdot 1!} = \frac{120}{4 \cdot 1} = 30 \).

Thus, the total number of ways to distribute the objects is \( 3 \cdot 30 = 90 \).

In summary, the approach involves determining how many objects each person will receive based on the total number of objects and people, and then applying combinatorial principles to count the distinct distributions. This method ensures that the distribution meets the condition of having a maximum difference of at most 1 in the number of objects received by any two individuals.