$CABBAGE$

I will show a simple method to solve problems like this one. I know that it is a combinatorics problem but we can use the principle of numerical bases to solve it.

Let's attribute a number to each alphabet of the set:

$A=0$

$B=1$

$C=2$

$E=3$

$G=4$

Now we can convert the "words list" to a numerical list:

$AAAAAAA=0000000$

$AAAAAAB=0000001$

$AAAAAAC=0000002$

$AAAAAAE=0000003$

$...$

$CABBAGE=2011043$

Notice that these numbers are in base 5 . Now we can convert CABBAGE to decimal:

$2\cdot5^6+0\cdot5^5+1\cdot5^4+1\cdot5^3+0\cdot5^2+4\cdot5^1+3\cdot5^0=31250+625+125+20+3=32023$

In this list AAAAAAA is in the "0th" position because it corresponds to the number 0. But it is in actually the 1st. So, CABBAGE is in the $32023+1=32024th$ position in this list.