Mistakes Give Rise to Problems- 18

In Computer Science feed, if you come across \(\color{Blue}{\mathrm{A+B+C+D+E=15}}\) and then asked to find the value of \(F+G+H+I+J\) , then one of the approaches is \[\underbrace{A+B+C+D+E=15}_{\text{1+2+3+4+5=15}}\] and then you assign the numbers \(F,G,H,I,J\) their respective numbers from position in Alphabets, giving \(\underbrace{F+G+H+I+J=40}_\text{6+7+8+9+10=40}\)

But if you do that in \(\textbf{Maths}\) , for some unknown positive integers \(A,B,C,D,E,F,G,H,I,J\) ,

\(A+B+C+D+E=15 \implies F+G+H+I+J=40\)

then it is a \(\color{Red}{\textbf{Big Mistake !!!}}\) (You can't say 1st equation implies the 2nd)

But for how many ordered 10-tuples of distinct positive integers \((A,B,C,D,E,F,G,H,I,J)\), are the above two equations simultaneously true ?

Details and assumptions :-

\(\bullet\) All the alphabets \(A,B,C,D,E,F,G,H,I,J\) are distinct and are some positive integers.

\(\bullet\) The tuple \((1,2,3,4,5,6,7,8,9,10)\) is different from \((2,1,3,4,5,6,7,8,9,10)\).

This is a part of the set Mistakes Give Rise to Problems!