Allow me to ask (may be a silly) question, I got my school exam tommorow and I'm stuck in a question. (Orignal question is different)
There is a bag with 9 balls including 3 blue , 2 red , 4 black balls. Balls of same color are non identical, we have to select 4 balls such that there must be atleast a ball of each color. I deduce answer from original question to be 144. And I did get 144 It went like Selecting one ball of each color amd then selecting 1 ball from left 6 balls. $$^4 C_1 × ^3 C_1× ^2 C_1× ^6 C_1$$ which gives 144. If the above math doesn't look the way I expected, then it is 4C1 × 3C1× 2C1 ×6C1
But, what confuses me is, I made 3 cases. First selected 2 balls from blue , one each from rest two, then did the same 3 times with other colors. Which added upto 72.
Answer should be 144. I don't know where I am lacking, is it a conceptual error?