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Step-by-Step Solution
Step 1: Understand the Problem
We have 3 sections in a question paper, each containing 5 questions. The candidate must answer exactly 5 questions in total, with at least one question from each section. We need to find the total number of ways to choose these questions.
Step 2: Possible Distributions of Questions
Let $x_1, x_2, x_3$ represent the number of questions chosen from sections 1, 2, and 3 respectively. Then:
$x_1 + x_2 + x_3 = 5$
with each $x_i \ge 1$ (because at least one question must come from each section) and $x_i \le 5$ (each section has 5 questions). The valid distributions are:
(1, 1, 3)
(1, 3, 1)
(3, 1, 1)
(1, 2, 2)
(2, 1, 2)
(2, 2, 1)
Step 3: Number of Ways for Distribution (1, 1, 3)
Choose 1 question from section 1, 1 from section 2, and 3 from section 3:
Number of ways = $C(5,1) \times C(5,1) \times C(5,3)$
where $C(n, r)$ denotes the combination function "n choose r". We have $C(5,1) = 5$ and $C(5,3) = 10$. So:
$C(5,1) \times C(5,1) \times C(5,3) = 5 \times 5 \times 10 = 250$
Since the "3" can appear in any one of the three sections, there are 3 such permutations (sections could be (3,1,1), (1,3,1), or (1,1,3)). Hence:
Total for patterns like (1,1,3) = $250 \times 3 = 750$
Step 4: Number of Ways for Distribution (1, 2, 2)
Choose 1 question from one section and 2 questions from each of the other two sections:
Number of ways for a specific arrangement (e.g., (1,2,2)):
$C(5,1) \times C(5,2) \times C(5,2)$
where $C(5,1) = 5$ and $C(5,2) = 10$. Therefore:
$5 \times 10 \times 10 = 500$
There are 3 ways to place the "1" among the three sections (that is, (1,2,2), (2,1,2), or (2,2,1)). Thus:
Total for patterns like (1,2,2) = $500 \times 3 = 1500$
Step 5: Sum of All Possible Ways
Adding the number of ways from both types of distributions:
$750 + 1500 = 2250$
Step 6: Conclusion
The total number of ways to choose 5 questions, with at least one from each section, is 2250.
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