Max (Six + Ten)

We can write the given summation as $100(S+T)+10(I+E)+X+N$ .

In order to make the above summation larger , we must maximize $(S+T)$ first. So we have $(S,T)=(8,9),(9,8)$ . Now we must maximize $(I+E)$ and we can do it by choosing the largest numbers before $8,9$ that are $6,7$ . Hence we have $(I,E)=(6,7),(7,6)$ . Now for maximizing $(X+N)$ we have $(X,N)=(4,5),(5,4)$ .

So the maximum summation possible is :

$975+864=874+965=974+865=964+875=\huge\boxed{1839}$

Since $1 \leq 1 \leq 10 \leq 10 \leq 100 \leq 100$ and $4 \leq 5 \leq 6 \leq 7 \leq 8 \leq 9$ then by the rearrangement inequality, The maximum value will be $100 \times 9 + 100 \times 8 + 10 \times 7 + 10 \times 6 + 5 + 4 = 1839$ Assigning $S = 9, T = 8, I = 7, E = 6, X = 5, N = 4$ gives this value.

Its simple, for this number to be greatest hundreds digits should have the maximum values. Assign highest possible values to the hundreds digits, $S$ and $T$ , i.e. $9$ and $8$ , then go on decreasing the values assign $7$ and $6$ to $I$ and $E$ , and $5$ and $4$ to $X$ and $N$ . Put the values and get the answer i.e. $1839$ .

Each letter has to be another number. The 100's have the highest value, so you use the highest value numbers for those. 975+864=1834 --> highest value.

Since each variable is a different letter, we must assume they r all different. Using the highest possible combination of numbers we place the highest valued numbers in the hundreds place, (S)=9 and (T)=8 . The next highest go in the tens place, (I)=7 and (E)=6, then (X)=5 and (N)=4 go in the ones place. Then it's simple addition, 5+4=9, 7+6=13, carry the one which makes 10+8=18. The answer becomes 1,839.

Starting from 9 start distributing the numbers in the hundreds, tens and the units.

A possible distribution is 976 + 864 = $\boxed{ 1030 }$