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Type Ia supernovae (SNe Ia) are thought to be the result of thermonuclear explosions in white dwarfs (WDs). Commonly considered formation pathways include two merging WDs (the double degenerate channel), and a single WD accreting material from a H or He donor (the single degenerate channel). Since the predicted SNe Ia rates from WD in binaries are thought to be insufficient to explain the observed SNe Ia rate, it is important to study similar interactions in higher-order multiple star systems such as triple systems. We use the evolutionary population synthesis code Multiple Stellar Evolution (MSE) to study stellar evolution, binary interactions and gravitational dynamics of the triple-star systems. Also, unlike previous studies, prescriptions are included to simultaneously take into account the single and double degenerate channels, and we consider triples across the entire parameter space (including those with tight inner binaries). We explore the impact of typically ignored or uncertain physics such as fly-bys and CE prescription parameters on our results. The majority of systems undergo circular mergers to explode as SNe Ia, while eccentric collisions contribute to $0.4-4$ per cent of SNe Ia events. The time-integrated SNe Ia rate from the triple channel is found to be $(3.60 \pm 0.04) \times {10^{-4}}\,\mathrm{M_{\odot}}^{-1}$ which is, surprisingly, similar to that of the isolated binary channel where the SNe Ia rate is $(3.2 \pm 0.1) \times {10^{-4}}\,\mathrm{M_{\odot}}^{-1}$. This implies that triples, when considering their entire parameter space, yield an important contribution to the overall SNe Ia rate.