How many solutions possible for the equation x₁+x₂+x3=11 where x ᵢ ≥0?
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How many solutions possible for the equation x₁+x₂+x3=11 where x ᵢ ≥0?
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Answer: B. 78
This is a stars and bars combinatorics problem. The number of non-negative integer solutions to x₁ + x₂ + x₃ = 11 is given by C(n + k - 1, k - 1) = C(13, 2) = 78. The other options are common distractors using incorrect formulas like C(11, 2) = 55 or permutations.
Step-by-step Derivation:
Using stars and bars formula:
- n = 11 (the sum we want)
- k = 3 (number of variables)
- Number of solutions = C(n + k - 1, k - 1) = C(11 + 3 - 1, 3 - 1) = C(13, 2)
- C(13, 2) = 13! / (2! × 11!) = (13 × 12) / (2 × 1) = 156 / 2 = 78