# Dirichlet box principle

In a more quantified version: for natural numbers $$k$$ and $$m$$, if $$n = km + 1$$ objects are distributed among $$m$$ sets, then the pigeonhole principle asserts that atleast one of the sets will contain at least $$k + 1$$ objects. (from wikipedia) Now why is 'atleast' there? Only one set should contain only $$k+1$$ objects, as $$n$$ is just one more than $$k$$ items (are being contained by $$m$$ sets) , i.e. $$n=km+1$$

• Not following. Maybe one set contains all the objects and all the others are empty. Or, if that example is missing your point, can you give a numerical example of what is bothering you?
– lulu
May 12, 2018 at 18:19
• No I am not able to grasp the actual essence of that line. Can you please give me example. On wikipedia there is example of 10 pigeons 9 holes so that is satisfied but any examples in favor of atleast May 12, 2018 at 18:23
• An example of what? I don't understand. If I have $2$ objects and $1$ hole then exactly $1$ of the holes has more than $1$ object. Is that an example of what you want?
– lulu
May 12, 2018 at 18:32
• Thank you Sir for your help, I have got my answer what I wanted. May 12, 2018 at 18:45

Let $k=3$ and $m=2$ so that $n=7$. Then one possible arrangement is $(5,2)$, but neither of these is $k+1=4$.