Use inclusion-exclusion:
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- Total sequences without restriction: $ 4^6 $
- Subtract sequences missing at least one type:
- Let $ S = 4^6 $
- ext{Number of surjective sequences} = \sum_{k=0}^{4} (-1)^k inom{4}{k} (4 - k)^6
- $ k = 0 $: $ (-1)^0 inom{4}{0} \cdot 4^6 = 1 \cdot 1 \cdot 4096 = 4096 $
- $ k = 1 $: $ (-1)^1 inom{4}{1} \cdot 3^6 = -1 \cdot 4 \cdot 729 = -2916 $
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