= Solution
{c}
Set $S=\sum_i x_i$ and $N=nm$. Conditional <independence> of the <binomial distribution> observations gives the <likelihood function>
$$
L(\theta)=\left\{\prod_{i=1}^n\binom m{x_i}\right\}
\theta^S(1-\theta)^{N-S}.
$$
The <prior distribution> is the <beta distribution> with parameters $(1,\beta)$. By <Bayes' theorem>, multiplying the <prior density> by the <likelihood function> gives a <posterior density> proportional to
$$
\theta^S(1-\theta)^{N-S+\beta-1}.
$$
Recognizing and normalizing this <beta distribution> gives
$$
\boxed{\theta\mid\boldsymbol x\sim\operatorname{Beta}(S+1,N-S+\beta)},\qquad
\boxed{\pi(\theta\mid\boldsymbol x)=
\frac{\theta^S(1-\theta)^{N-S+\beta-1}}{B(S+1,N-S+\beta)}}
$$
for $0<\theta<1$. This is <Beta-binomial conjugacy>; the original binomial coefficients cancel in the normalization because they are independent of $\theta$.
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