How many integer solutions (x_1, x_2, x_3) satisfy x_1 + x_2 + x_3 = 12, where x_1 >= 1, x_2 >= 1, and x_3 >= 1?
Answer and explanation
Correct answer: A. 55
- Step 1: Use the stars and bars method for positive integer solutions.
- Step 2: Formula: C(n - 1, k - 1), where n = 12 and k = 3.
- Step 3: C(12 - 1, 3 - 1) = C(11, 2) = (11 * 10) / 2 = 55.
Answer
55.
Why not the others: 66 computes C(12, 2); 78 computes C(13, 2); 91 computes C(14, 2) (which applies for non-negative solutions x_i >= 0).
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