I am a biology student and i am stuck with a problem, hence here. I have a large number of 10 different kinds of viruses in a tube and i need to infect cells with it. i need to do the infection in a way that each cell gets only one virus (and this is not the problem). i know that if i just infect 100 cell, the likelihood of getting at least one virus of each kind is not great. Instead if i increase the number of infected cells to say 1000 and each cell is infected by only 1 virus, then the likelihood of getting at least one virus of each kind increases. I am wondering if there is a formula to show this phenomenon and maybe a graphical way to illustrate that increasing the number of infected cells, increases chances of getting each virus at least once? I think a way to generalise my problem is: i have loads of 10 different colored balls and i need to put them into boxes. All boxes are identical and each box can contain only one ball. Increasing the number of boxes with balls at the end of the experiment, i can ensure that at least there will be one ball of each color in a box. I know bits of R so it wil be helpful if i could create some graphs in R. Can anyone give any input on this? Any help is much appreciated

Thanks very much