Binet's Formula Proof (Einstein notation, Levi-Civita symbol)

Question

I'm trying to understand the proof below... Although I'm familiar with the notation itself, I'm in trouble with the underlined statement... can anyone explain?

Answer 1

This is a bit confusing offhand because, after all, in the original sum $l_1,\dots,l_n$ needn't even be distinct. But let's consider a simpler example. Take $n=2$ and let's look at two cases. When $l_1=l_2=1$, $$b_{l_1k_1}b_{l_2k_2}\epsilon_{k_1k_2} = b_{11}b_{12} - b_{12}b_{11} = 0;$$ the terms cancel because we can interchange $k_1$ and $k_2$ without changing the product. When $l_1$ and $l_2$ are distinct, we have either $$b_{1k_1}b_{2k_2}\epsilon_{k_1k_2}\quad\text{or}\quad b_{2k_1}b_{1k_2}\epsilon_{k_1k_2};$$ to turn the second sum into the first, we must interchange $k_1$ and $k_2$, so these sums differ by a factor $-1 = \epsilon_{21}$.

The general case is conceptually the same, just a bit messier to say. You first show that when any of the $l_i$ duplicate, the sum is $0$. You next show that when you permute $l_1,\dots,l_n$, the sums differ by the sign of that permutation.