1. **Problem statement:** A student must answer 10 out of 13 questions. (i) He must answer at least the first two from the first 5 questions. 2. **Understanding the problem:** - Total questions: 13 - Must answer: 10 - Condition: Must answer questions 1 and 2 (at least the first two from the first 5) 3. **Approach:** - Since questions 1 and 2 must be answered, these 2 are fixed. - Remaining questions to answer: $10 - 2 = 8$ - Remaining questions available: $13 - 2 = 11$ 4. **Calculate the number of ways:** - The student can choose any 8 questions from the remaining 11 questions. - Number of ways is given by the combination formula: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ - Here, $n=11$, $k=8$ 5. **Calculate $\binom{11}{8}$:** $$\binom{11}{8} = \binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165$$ 6. **Final answer:** - The student can answer the questions in **165** different ways under the given condition.