I was studying Permutation in a 'Groups' class. They thought me that the multiplication of two tuples $\alpha\beta=(1\,2\,3)(2\,4\,1\,3)$ in $S_5$ is defined as following:
$$ \alpha\beta(1)=\alpha3=1\\ \alpha\beta(2)=\alpha4=4\\ \alpha\beta(3)=\alpha2=3\\ \alpha\beta(4)=\alpha1=2\\ \alpha\beta(5)=\alpha5=5$$
So we get: $\alpha\beta=(1\,2\,3)(2\,4\,1\,3)=(2\,4)$ But from other books, they calculate from left to right:
$$ (1)\alpha\beta=2\beta=4\\ (2)\alpha\beta=3\beta=2\\ (3)\alpha\beta=1\beta=3\\ (4)\alpha\beta=4\beta=1\\ (5)\alpha\beta=5\beta=5$$
So we get: $\alpha\beta=(1\,2\,3)(2\,4\,1\,3)=(1\,4)$
I do understand that both ways are not equal but I don't understand when do I use each way. Asked my teacher and he replied that we use it as in first way because we speak about functions so if $fg$ then $f(g(x))$. But then why other books use the second way? Also tried to use Wolfram (link) and it gave the second result.