两只繁殖场狗狗如何度过她的九年
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\begin{aligned} \sum_{i = 1}^{2^n - 1} \binom{n}{f_i} &= \sum_{i = 1}^{n} \binom{n}{i} 2^{n - i} \\ &= \sum_{i = 1}^{n} \binom{n}{i}2^{n - i} \\ &组合意义，3种颜色染色。 \\ &= 3^{n} - 2^{n} \end{aligned}
i=1∑2n−1(fin)=i=1∑n(in)2n−i=i=1∑n(in)2n−i组合意义，3种颜色染色。=3n−2n
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\begin{aligned} \sum_{i = 1}^{n} \binom{n}{i}2^{n - i} &= \sum_{i = 0}^{n - 1} \binom{n}{i} 2^i \\ &= -2^{n} + \sum_{i = 0}^{n} \binom{n}{i}2^i \\ &= -2^{n} + (2+1)^{n} \\ &= -2^{n} + 3^{n} \\ &= 3^n - 2^n \end{aligned}
i=1∑n(in)2n−i=i=0∑n−1(in)2i=−2n+i=0∑n(in)2i=−2n+(2+1)n=−2n+3n=3n−2n
∑ i = 1 2 n − 1 g i f i ( n f i ) = ∑ i = 1 n 2 n − i ( 2 i − 1 ) ( n i ) i = ∑ i = 1 n \begin{aligned} \sum_{i = 1}^{2^n - 1} g_i f_i \binom{n}{f_i} &= \sum_{i = 1}^{n} 2^{n - i} (2^{i} - 1) \binom{n}{i} i \\ &= \sum_{i = 1}^{n} \end{aligned} i=1∑2n−1gifi(fin)=i=1∑n2n−i(2i−1)(in)i=i=1∑n
