= Solution
{c}
Let $s=\sum_i x_i$. The <Bernoulli distribution> <likelihood function> is $\theta^s(1-\theta)^{n-s}$. Multiplication by the <beta distribution> <prior density> and <Bayes' theorem> give
$$
\pi(\theta\mid x)\propto\theta^{a+s-1}(1-\theta)^{b+n-s-1},\qquad0<\theta<1.
$$
Recognizing the normalized <beta distribution>,
$$
\boxed{\theta\mid x\sim\operatorname{Beta}(a+s,b+n-s),\qquad
\pi(\theta\mid x)=\frac{\theta^{a+s-1}(1-\theta)^{b+n-s-1}}{B(a+s,b+n-s)}.}
$$
This is <Beta-binomial conjugacy>; the same update applies whether the data are recorded as the individual <Bernoulli trials> or their success count.
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