Question: In (mn+1) pairs of numbers are written down, each pair consisting of one chosen from the m letters $a_{1},a_{2},a_{3},.........,a_{m},$ and one from the n letters $b_{1},b_{2},b_{3},.........,b_{n}$, prove that at least two of the pairs are identical.

Please can some one show how to do this question I dont know how to tackle this question

how many unique pairs $(a_j,~b_k),~1\leq j \leq m,~1 \leq k \leq n$ can be formed?
Question: In (mn+1) pairs of numbers are written down, each pair consisting of one chosen from the m letters $a_{1},a_{2},a_{3},.........,a_{m},$ and one from the n letters $b_{1},b_{2},b_{3},.........,b_{n}$, prove that at least two of the pairs are identical.
If $A=\{a_k:1\le k\le m\}~\&~B=\{b_k:1\le k\le n\}$ then what is $A\times B$ ? Is it a set of ordered pairs?
If so how many pairs are there in $A\times B~?$ Among $mn+1$ pairs taken from $A\times B$ are there two identical? WHY?