Calculate cardinality of a set

Question

There is given:

• set $B=A_{1} \cup...\cup A_{n}$
• $|A_{i}|=m_{i}$ for each $i$
• every element of $B$ belongs to exactly $k$ sets $A_{i_{1}},...,A_{i_{k}}$

Calculate $|B|$ in terms of $m_{i}$. If it has not unique solution then find its upper bound.

I tried to use inclusion-exlusion principle but so far my estimation is too big. It is $m_{1}+...+m_{n}$

Is there any better estimations?

Regards.

Answer 1

Every element of $B$ belongs to exactly $k$ sets out of the $A_i$. So if we count all the elements of the $A_i$, we have counted each element of $B$ exactly $k$ times. This means that $$|A_1|+|A_2|+\cdots+|A_n|=k|B|.$$