Mistakes give rise to Problems- 14

\[\color{Red}{\textbf{JOMO 7, Short 8}}\]

There are two sequences of integers {\(a_i\)}\(_{i=1}^4\) and {\(b_i\)}\(_{i=1}^4\)

If you do as what is shown, if you cancel the \(\sum\) sign, like \[\displaystyle \color{Blue}{\dfrac{\sum_{j=1}^4 a_j}{\sum_{j=1}^4 b_j }= \dfrac{a_i}{b_i}} \quad \quad \quad \forall i \in \{1,2,3,4 \}\] then it will be a \(\color{Red}{\textbf{Big Mistake !!!}}\).

But for how many ordered pairs of sequences (<\(a_i\)> , <\(b_i\)>) such that \[1 \leq a_1 < a_2 < a_3 < a_4 \leq 20\] \[1 \leq b_1 < b_2 < b_3 < b_4 \leq 20\] is the above said "False" property seen to be "True" ?

\(\bullet\) This problem is a part of the set Mistakes Give Rise To Problems , and this question appeared in JOMO 's Contest #7 , and was posed by me.