Summary: For positive integers \(n \geq m\), let \(P_{n, m}(x) := \sum_{j = 0}^{m} \binom{n}{j} x^{j} = \binom{n}{0} + \binom{n}{1}x + \dots + \binom{n}{m}x^{m}\) be the truncated binomial expansion of \((1 + x)^{n}\) consisting of all terms of degree \(\leq m\). It is conjectured that for \(n > m + 1\), the polynomial \(P_{n, m}(x)\) is irreducible. We confirm this conjecture when \(2m \leq n < (m + 1)^{10}\). Also we show for any \(r \geq 10\) and \(2m \leq n < (m + 1)^{r + 1}\), the polynomial \(P_{n, m}(x)\) is irreducible when \(m \geq \max\{10^{6}, 2r^{3}\}\). Under the explicit abc-conjecture, for a fixed \(m\), we give an explicit \(n_{0}\), \(n_{1}\) depending only on \(m\) such that \(\forall n \geq n_{0}\), the polynomial \(P_{n, m}(x)\) is irreducible. Further \(\forall n \geq n_{1}\), the Galois group associated to \(P_{n, m}(x)\) is the symmetric group \(S_{m}\).