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But we must subtract or eliminate sequences where **identical nucleotides are adjacent**, particularly the two A’s and the three C’s or three G’s.
However, due to the complexity of inclusion-exclusion over multiple repeated elements and spatial constraints, we instead model this using **inclusion with arrangement constraints** — specifically, placing the letters so that no two identical ones are adjacent.
A better approach is constructive: build the sequence step-by-step, placing the letters while avoiding adjacent duplicates.
But because we have repeated letters and exact counts, we use a **recursive or combinatorial placement strategy** with the principle of **arrangements with no adjacent duplicates** for multiset permutations.
However, due to the complexity and size, we apply a known combinatorial method: **counting valid permutations with forbidden adjacency** using inclusion-exclusion or generating functions — but a more efficient way is to use **placement algorithms**.
Alternatively, we proceed by **constructing valid sequences** using dynamic programming or recursive counting, but for Olympiad-style reasoning, we can use the **principle of arranging with separation**.
But here’s a better and feasible approach:
We use **inclusion-exclusion over forbidden adjacents**, but that becomes intractable due to multiple overlapping cases.
Instead, we use a **constructive counting method** with **placement strategy**.
Let’s consider that we are placing 8 positions, assigning A, C, G with multiplicities 2, 3, 3, such that **no two identical letters are adjacent**.