= Solution
Under a <risk-neutral probability in a binomial market>[risk-neutral measure], the discounted stock must be a <martingale>. If $q$ is the probability of the factor $1+b$, this condition is
$$
q(1+b)+(1-q)(1+a)=1+r,
$$
so
$$
\boxed{q=\frac{r-a}{b-a}}.
$$
The hypothesis $a<r<b$ ensures $0<q<1$, making this equivalent to the original probability measure. Since the $N$ returns are independent under $Q$, the number of up moves has a <binomial distribution>, and hence
$$
\boxed{
Q\!\left(S_N=S_0(1+b)^i(1+a)^{N-i}\right)
=\binom Niq^i(1-q)^{N-i}}.
$$
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