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(A) The question paper has $12$ questions: Part $I$ ($5$ questions) and Part $II$ ($7$ questions).$A$ student must select $8$ questions in total,with

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$420$

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(A) The question paper has $12$ questions: Part $I$ ($5$ questions) and Part $II$ ($7$ questions).$A$ student must select $8$ questions in total,with at least $3$ from each part. The possible combinations are:$(a)$ $3$ questions from Part $I$ and $5$ questions from Part $II$: $^{5}C_{3} 	imes ^{7}C_{5} = 10 	imes 21 = 210$ ways.$(b)$ $4$ questions from Part $I$ and $4$ questions from Part $II$: $^{5}C_{4} 	imes ^{7}C_{4} = 5 	imes 35 = 175$ ways.$(c)$ $5$ questions from Part $I$ and $3$ questions from Part $II$: $^{5}C_{5} 	imes ^{7}C_{3} = 1 	imes 35 = 35$ ways.Total ways = $210 + 175 + 35 = 420$ ways.

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