A verification of Pigeon hole Principle I was wondering that the generalization of Pigeon hole Principle states that :

If $kn+1$ pigeons are distributed in $n$ holes, then some  hole has atleast $k+1$ pigeons.

This is quite obvious intuitively however, if we consider that amongst let's say $nk$ pigeons of $nk+1$ of them are considered and  those $nk$ pigeons are first distributed in $n $ holes. So, if some holes has $k$ pigeons and some of them has less than $k$ pigeons ...Then is this situation possible as then we can keep the $1$ remaining pigeon in the hole containing less than $k$ pigeons and then all the $n$ holes has less than k+1 pigeons ...then it doesn't verify the Pigeon hole principle stated above...where is the problem occurring?
 A: Move the pigeons around
Here's one way to put $kn$ pigeons into $n$ holes: put $k$ pigeons into each hole so that each hole has the same number.
Now if you want to put the pigeons into the holes in a different way, you can just move these pigeons around.  But if you move any pigeons around, notice that some of the holes will have more than $k$ pigeons and some of the holes will have less than $k$ pigeons. The only way one of the holes can have less than $k$ pigeons in it is if another hole has more than $k$ pigeons in it.
So: when you put $nk$ pigeons into $n$ holes, either

*

*Some of the holes have more than $k$ pigeons and some have less than $k$ pigeons. Then the generalized pigeonhole principle is already established in this case.


*All of the holes have exactly the same number of pigeons, $k$. Adding one more pigeon to any hole will establish the generalized pigeonhole principle in this case.

Or notice that there aren't enough pigeons
First you put $kn$ pigeons into $n$ holes, then you'll decide what to do with the last pigeon.
If any of the holes has more than $k$ pigeons in it already, you're done—you've proved the generalized pigeonhole principle for this case.
Otherwise, all of the holes have $k$ pigeons or fewer. But remember we said the total number of pigeons must be $kn$.  Note that in order to add up to the correct total, every hole must have exactly $k$ pigeons in it. It's impossible for any hole to have less than $k$ pigeons in it in this case.
Why? If even one hole has $k-1$ pigeons in it and the other holes all have $k$ pigeons, then the total number of pigeons is $kn-1$—too small. We're missing some pigeons.
So when you say:

if some holes has $k$ pigeons and some of them has less than $k$ pigeons

This is where the mistake is— you imagine that you can have less than $k$ pigeons in a hole when this is impossible. If you put $nk$ pigeons into $n$ holes, and no hole has more than $k$ pigeons, then no hole has less than $k$ pigeons, either.
A: If for $i=1,\ldots,n$ there are $p_i$ pigeons in hole $i$ and if we assume that $p_i\le k$, then the total number $p$ of pigeons is
$$ p=p_1+p_2+\cdots +p_n\le k+k+\cdots +k=nk.$$
Hence,

If all $p_i$ are $\le k$, then $p\le nk$.

which is equivalent to

If not $p\le nk$, then not all $p_i$ are $\le k$.

or,

If $p>nk$, then there is some $i$ with $p_i>k$.

