= Solution
For the <European call option>, risk-neutral valuation gives
$$
\begin{aligned}
x
&=(1+r)^{-N}\mathbb E_Q[(S_N-K)^+]\\
&=(1+r)^{-N}\mathbb E_Q[S_N\mathbf1_{\{S_N>K\}}]
-K(1+r)^{-N}Q(S_N>K).
\end{aligned}
$$
Define the <stock-numeraire measure in a binomial market> by
$$
\widehat Q(A)
=\frac{\mathbb E_Q[S_N\mathbf1_A]}{S_0(1+r)^N}.
$$
It is a probability measure because the risk-neutral martingale property gives $\mathbb E_QS_N=S_0(1+r)^N$. Consequently
$$
\boxed{
x=S_0\widehat Q(S_N>K)
-K(1+r)^{-N}Q(S_N>K)}.
$$
Tilting one up move by its stock factor changes its probability to
$$
\boxed{
\widehat q=\frac{q(1+b)}{1+r}
=\frac{(1+b)(r-a)}{(1+r)(b-a)}}.
$$
Similarly $1-\widehat q=(1-q)(1+a)/(1+r)$. Thus under $\widehat Q$ the number of up moves is binomial with parameter $\widehat q$, and
$$
\boxed{\widehat Q\!\left(S_N=S_0(1+b)^i(1+a)^{N-i}\right)
=\binom Ni\widehat q^{\,i}(1-\widehat q)^{N-i}.}
$$
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