The number of ways to distribute 10 identical red pens and 14 identical blue pens among four persons such that each person gets 6 pens is __________.
Step-by-Step Solution
Key Concept: When distributing identical objects with both a minimum (0) and maximum (6) per person, stars-and-bars gives the unconstrained count, and inclusion-exclusion removes cases where any variable violates the upper bound.
**Step 1: Set up variables**
Let person $i$ receive $r_i$ red and $b_i = 6 - r_i$ blue pens. Constraints: $r_i \in \{0,1,\ldots,6\}$ and $\sum_{i=1}^{4} r_i = 10$.
**Step 2: Count solutions by inclusion-exclusion**
Without upper bound: $\binom{13}{3} = 286$. Subtract cases with some $r_i \ge 7$: set $r_i' = r_i-7$, then $\sum r_i'=3$, giving $\binom{4}{1}\binom{6}{3} = 80$. No two $r_i \ge 7$ simultaneously (sum would exceed 10). Valid: $286 - 80 = \mathbf{206}$.
Correct Answer: