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Contribution

The second term is the expected value of the first when we randomized the recipes.

We go through everything

$$\sum_{N_c}$$

The case where the recipe has ingredient $i$

$$\sum_{N_c} \frac{f_i}{N_c}$$

We need the same combination term

$$\sum_{N_c} \frac{f_i}{N_c} \frac{2}{n_R (n_R – 1)}$$

But we can replace with the expected value

$$\sum_{N_c} \frac{f_i}{N_c} \frac{2}{\langle n_R \rangle (\langle n_R \rangle – 1)}$$

Now we are thinking about the rest of the ingredient in this particular recipe ($\langle n_R \rangle – 1$)

$$\sum_{N_c} \frac{f_i}{N_c} \frac{2}{\langle n_R \rangle (\langle n_R \rangle – 1)} (\langle n_R \rangle – 1)$$

And so on…