I'm trying to build up my intuition about likelihood and came up with the most general problem I could think up.
Suppose $m$ people flip an unfair coin $k$ times and $Z_i$ is the number of people who landed $i$ heads where $i\in{0,1,...,k}$. Would the likelihood $l(Z_0,Z_1,...,Z_k|p)$ be computed as:
$l(Z_0,Z_1,...,Z_k|p)=\Pi_{i=0}^k l(Z_i|p)=\Pi_{i=0}^k {km \choose iZ_i} p^{iZ_i}(1-p)^{km - iZ_i}$
Is this correct?
There should be a multinomial factor in front representing the permutations of the people who got the various numbers of heads. Then the per-factor exponent sum of $p$ and $1-p$ should equal $Z_ik$ since that is the total number of coins flipped by the relevant people. The correct likelihood is $$\binom m{Z_0,\dots,Z_k}\prod_{i=0}^k\left(\binom kip^i(1-p)^{k-i}\right)^{Z_i}$$