A question paper is set in two parts: Part A carries 7 questions and Part B carries 5. A candidate has to attempt 6 questions in all, taking at least 2 from each part. In how many ways can the candidate choose the questions?

Split by how many come from Part A. 2 from A and 4 from B: $C(7,2)\times C(5,4)=21\times 5=105$. 3 and 3: $C(7,3)\times C(5,3)=35\times 10=350$. 4 and 2: $C(7,4)\times C(5,2)=35\times 10=350$. Taking 5 or 6 from Part A would leave fewer than 2 from Part B, so those splits are out. Total = 105 + 350 + 350 = 805. Trap $924=C(12,6)$ ignores the two-from-each rule. Traps 700 and 455 each drop one of the three valid splits.