Permutations and Combinations
Stars and Bars
Premium Question
Grade 11
Question:
The number of ways to distribute $12$ identical balls into $4$ distinct boxes such that each box contains at least $2$ balls is:
(1) $15$
(2) $35$
(3) $55$
(4) $20$
Step-by-Step Solution
Key Concept: Substitute $x_i = y_i + 2$ (each $y_i \geq 0$): $y_1 +y_2 +y_3 +y_4 = 4$. Stars and bars: $\binom{4+3}{3} = \binom{7}{3} = 35$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)