# 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!

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