By employing one of the cubic transformations (due to W. N. Bailey (1928)) for the ${}_3{F}_2\left(x\right)$-series, we examine a class of ${}_3{F}_2\left(4\right)$-series. Several closed formulae are established by means of differentiation, integration and contiguous relations. As applications, some remarkable binomial sums are explicitly evaluated, including one proposed recently as an open problem.