Expansions are provided for the moments of the number of collisions $X_n$ in the $β(2, b)$-coalescent restricted to the set $\{1, …, n\}$. We verify that $X_{n}/\mathbb{E}X_{n}$ converges almost surely to one and that $X_n$, properly normalized, weakly converges to the standard normal law. These results complement previously known facts concerning the number of collisions in $β(a, b)$-coalescents with $a∈(0, 2)$ and $b=1$, and $a>2$ and $b>0$. The case $a=2$ is a kind of ‘border situation’ which seems not to be amenable to approaches used for $a≠2$.