$ k = 5 $: $ -\binom{6}{5} \cdot 1^{10} = -6 \cdot 1 = -6 $
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- $ k = 2 $: $ \binom{6}{2} \cdot 4^{10} = 15 \cdot 1048576 = 15728640 $
- $ k = 3 $: $ -\binom{6}{3} \cdot 3^{10} = -20 \cdot 59049 = -1180980 $
- $ k = 4 $: $ \binom{6}{4} \cdot 2^{10} = 15 \cdot 1024 = 15360 $
- $ k = 6 $: $ \binom{6}{6} \cdot 0^{10} = 1 \cdot 0 = 0 $
- Now sum:
- - 58593750 + 15728640 - 1180980 + 15360 - 6 = \boxed{2941680}
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