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 ________.

$10\ \text{red (identical)}, \quad 14\ \text{blue (identical)}$

$4\ \text{persons, each gets exactly } 6 \text{ pens}$

$\text{To find: the number of ways to distribute}$

$10 + 14 = 24 = 4 \times 6 \quad (\text{all pens are used})$

$\text{Key: fix the red counts } r_i; \text{ then blue is forced}$

$r_1 + r_2 + r_3 + r_4 = 10$

$b_i = 6 - r_i$

$b_i \ge 0 \Rightarrow r_i \le 6$

$0 \le r_i \le 6, \quad r_1 + r_2 + r_3 + r_4 = 10$

$\text{non-negative solutions} = \binom{10+3}{3} = \binom{13}{3}$

$\binom{13}{3} = \frac{13 \times 12 \times 11}{6} = 286$

$\text{Over-counted: need } r_i \le 6, \text{ so remove } r_i \ge 7$

$r_i' = r_i - 7 \ge 0$

$r_i' + \sum_{j \ne i} r_j = 10 - 7 = 3$

$\binom{3+3}{3} = \binom{6}{3} = 20$

$2 \times 7 = 14 > 10 \Rightarrow \text{at most one } r_i \ge 7$

$\text{Invalid} = 4 \times 20 = 80$

$N = \binom{13}{3} - 4\binom{6}{3} = 286 - 80 = 206$

$\boxed{N = 206}$