<s>[INST] How many possible words of this type can be formed? We are making $10$ letter words using the letters $A,C,G,T$.
How many possible words are there of the form $A...AC...CG...GT...T$ 
This is where all of the $A's$ go before the all of the $C's$ and all of the $C's$ go before all of the $G's$ and all of the $G's$ go before all of the $T's$?
I know there are $4^{10}$ possible words comprised of the letters $A,C,G,T$ but I have no idea how to account for the overcounting in this bit.
 [/INST] This problem is equivalent to finding the number of ways to partition $10$ into four parts, in which the order of the partitioning matters. For example, $10 = 2 + 3 + 3 + 2$ is not the same as $10 = 3 + 2 + 2 + 3$. The first number represents the number of A's, the second number represents the number of C's, and so on.
In order to then count the number of ways, think of $10$ dots and $3$ lines. The $3$ lines will divide the $10$ dots into four parts. So, we have a total of $10 + 3 = 13$ spaces to choose from, and we need to place 3 lines. Therefore, the answer will be ${13 \choose 3} = 286$.
Illustrations of $10$ dots and $3$ lines:
$$\cdot\cdot|\cdot\cdot\cdot|\cdot\cdot\cdot\cdot|\cdot$$
$$\cdot\cdot\cdot|\cdot|\cdot|\cdot\cdot\cdot\cdot\cdot$$
$$\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot|\cdot\cdot||$$
$$\cdot\cdot\cdot|\cdot\cdot\cdot|\cdot\cdot|\cdot\cdot$$
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