An exam paper has three sections, each holding 4 questions. A candidate must attempt exactly 5 questions in all and must attempt at least one question from each section. In how many ways can the five questions be chosen?

Five questions spread over three sections with at least one from each can only split as 3–1–1 or 2–2–1. Split 3–1–1: for a named section supplying the three, $C(4,3)\times C(4,1)\times C(4,1)=4\times 4\times 4=64$, and any of the 3 sections could be that one, giving 192. Split 2–2–1: for a named section supplying the single question, $C(4,2)\times C(4,2)\times C(4,1)=6\times 6\times 4=144$, and again 3 choices, giving 432. Total = 192 + 432 = 624. Trap $792=C(12,5)$ ignores the one-per-section rule. Traps 432 and 192 each report one of the two splits on its own.