Mistakes Give Rise to Problems- 18

Probability Level 4

In Computer Science feed, if you come across A+B+C+D+E=15\color{#3D99F6}{\mathrm{A+B+C+D+E=15}} and then asked to find the value of F+G+H+I+JF+G+H+I+J , then one of the approaches is A+B+C+D+E=151+2+3+4+5=15\underbrace{A+B+C+D+E=15}_{\text{1+2+3+4+5=15}} and then you assign the numbers F,G,H,I,JF,G,H,I,J their respective numbers from position in Alphabets, giving F+G+H+I+J=406+7+8+9+10=40\underbrace{F+G+H+I+J=40}_\text{6+7+8+9+10=40}

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

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

then it is a Big Mistake !!!\color{#D61F06}{\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)(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,JA,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)(1,2,3,4,5,6,7,8,9,10) is different from (2,1,3,4,5,6,7,8,9,10)(2,1,3,4,5,6,7,8,9,10).

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


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