I have $1000$ colors, and I want to divide these colors among $5$ people such that each color is given to at least $3$ people. How much should I give each user such that each color is given to at least $3$ people and each user gets the least number of colors while doing a fair distribution? Is there a general formula that I can use?
for example, if we have 8 colors: $1,2,3,4,5,6,7,8$ and $5$ users
user 1: 1,2,3,6,7,8
user 2: 3,4,5,1,2,3
user 3: 3,4,5,1,2,3
user 4: 6,7,8,3,4,5
user 5: 6,7,8,3,4,5
I am guessing that you want to distribute the minimum so that each color is distributed to at least $3$ people.
As has been commented, the problem is trivial (i.e. where you could give all $1000$ colors to persons 1 through 3, ignoring persons 4 and 5), unless you add the constraint that each person receives the same number of colors.
The following charted solution solves the problem, as I am interpreting it : each person gets the same number of colors.
Consider $~\displaystyle \binom{5}{3} = 10.$ This represents the number of ways of selecting $5$ items, without replacement, where order of selection is deemed irrelevant. These $10$ selections are represented in the chart below.
Consider for the moment, the simpler problem of $10$ colors, to be given to $5$ people, as charted below.
\begin{array}{| l | l | l | l | l | l |} \hline \text{Color} & \text{Person 1} & \text{Person 2} & \text{Person 3} & \text{Person 4} & \text{Person 5} \\ \hline 1 & x & x & x & & \\ \hline 2 & x & x & & x & \\ \hline 3 & x & x & & & x \\ \hline 4 & x & & x & x & \\ \hline 5 & x & & x & & x \\ \hline 6 & x & & & x & x \\ \hline 7 & & x & x & x & \\ \hline 8 & & x & x & & x \\ \hline 9 & & x & & x & x \\ \hline 10 & & & x & x & x \\ \hline \end{array}
In distributing $10$ colors to $5$ people, so that each color is distributed to at least $3$ people, you must make a minimum of $10 \times 3 = 30$ [color-person] distributions.
Therefore, the above chart represents the minimum, for $10$ colors.
Since $10$ is a factor of $1000$, you simply repeat the above chart $100$ times, to achieve the minimum [color:person] distribution.
For example, each of the following colors would be given specifically to persons $~1,~2~$ and $~3,~$ only:
color-1, color-11,color-21, ..., color-991.