# Mistakes Give Rise to Problems- 18

Probability Level 4

In Computer Science feed, if you come across $\color{#3D99F6}{\mathrm{A+B+C+D+E=15}}$ 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 $\color{#D61F06}{\textbf{Big Mistake !!!}}$ (You can't say 1st equation implies the 2nd)

But for how many ordered 10-tuples of distinct positive integers $(A,B,C,D,E,F,G,H,I,J)$, 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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