= Solution
A <European call option> with maturity $N$ and strike $K$ pays
$$
(S_N-K)^+
$$
at time $N$. Let $q$ be the risk-neutral conditional probability of the factor $1+b$. The discounted stock must be a martingale, so
$$
q(1+b)+(1-q)(1+a)=1+r,
$$
and therefore
$$
q=\frac{r-a}{b-a},
\qquad
1-q=\frac{b-r}{b-a}.
$$
Both probabilities are strictly positive. At every node these are the unique probabilities satisfying the martingale condition, so the <fundamental theorem of asset pricing> gives a unique no-arbitrage price.
If exactly $n$ of the $N$ moves use the factor $1+a$, then
$$
S_N=S_0(1+a)^n(1+b)^{N-n},
$$
and there are $\binom Nn$ such paths. Discounted risk-neutral expectation now gives
$$
\operatorname{EC}(N,K)
=\sum_{n=0}^N w(n,N)
\left(S_0(1+a)^n(1+b)^{N-n}-K\right)^+,
$$
where
$$
\boxed{
w(n,N)
=\frac1{(1+r)^N}\binom Nn
\left(\frac{b-r}{b-a}\right)^n
\left(\frac{r-a}{b-a}\right)^{N-n}}.
$$
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