Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are…

Let the four boxes be Box-1, Box-2, Box-3 and Box-4. Each of the box has $3$ red balls and $2$ blue balls.

$10$ balls can be chosen in:

Case I - When exactly one box gives four balls

Case II - When exactly two boxes gives three balls

Required number of ways$=$ Number of ways in Case I $+$ Number of ways in Case II.

$={}^{4}C_1\left[{}^{2}C_1+{}^{3}C_1\right]{\left({}^{3}C_1{}^{2}C_1\right)}^3+{}^{4}C_2{\left[{}^{3}C_2{}^{2}C_1+{}^{3}C_1{}^{2}C_2\right]}^2{\left[{}^{3}C_1{}^{2}C_1\right]}^2$

$=4\left(5\right){\left(6\right)}^3+6{\left(3\times 2+3\right)}^2{\left(6\right)}^2$

$=4320+17496=21816$