I am having trouble understanding where these last two expressions come from, namely:
$ P(S_n = i) = p^{\frac{n-j}{2} + \frac{i}{2}} q^{\frac{n-j}{2} - \frac{i}{2}} $
$ P(S_n = -i) = p^{\frac{n-j}{2} - \frac{i}{2}} q^{\frac{n-j}{2} + \frac{i}{2}} $
Shouldn't there be a binomial coefficient somewhere since we are calculating the number of successes in $n-j$ trials?
Example: let $j=0$, $n=10$, $i=4$.
Here event $S_{10} = 4$ could occur in multiple ways:
$S_1 = 1, S_2 = 2, \dots , S_7 = 7, S_8 = 6, S_9 = 5, S_{10} = 4$
OR
$S_1 = -1, S_2 = -2, S_3 = -3, S_4 = -2, S_5 = -1, \dots , S_9 = 3, S_{10} = 4$
So not only am I confused about where the probabilities come from but I have no intuition for it because there is no binomial coefficient present.