$ k = 1 $: $ (-1)^1 \binom{5}{1} \cdot 4^8 = -1 \cdot 5 \cdot 65536 = -327680 $
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- Then the number of onto functions is:
- \sum_{k=0}^{5} (-1)^k \binom{5}{k} (5 - k)^8
- $ k = 0 $: $ (-1)^0 \binom{5}{0} \cdot 5^8 = 1 \cdot 1 \cdot 390625 = 390625 $
- $ k = 2 $: $ (-1)^2 \binom{5}{2} \cdot 3^8 = 1 \cdot 10 \cdot 6561 = 65610 $
- $ k = 3 $: $ (-1)^3 \binom{5}{3} \cdot 2^8 = -1 \cdot 10 \cdot 256 = -2560 $
- $ k = 4 $: $ (-1)^4 \binom{5}{4} \cdot 1^8 = 1 \cdot 5 \cdot 1 = 5 $
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