# There are $11$ colored balls, $2$ are Red$, 2$ are Green$, 3$ are Yellow, and $4$ are Blue, select $n$ distinctly-colored balls.

Given a set of colored balls (multiple balls having the same color is allowed), is there any direct combinatorial approach to determine the total number of ways to select n distinctly-colored balls? Note: n will be less than or equal to the total number of distinct colors.

Example:- Let's say we have $$11$$ colored balls. $$2$$ are Red, $$2$$ are Green, $$3$$ are Yellow, and $$4$$ are Blue. Here, there are $$4$$ distinct colors. So n will be less then or equal to $$4.$$

Consider $$n = 3.$$ The possible combinations of selecting 3 distinctly-colored balls are {Red, Green, Yellow}, {Red, Green, Blue}, {Red, Yellow, Blue}, {Green, Yellow, Blue}.

Ways to select {Red, Green, Yellow} combination is $$2\cdot2\cdot3 = 12.$$

Ways to select {Red, Green, Blue} combination is $$2\cdot 2\cdot 4 = 16.$$

Ways to select {Red, Yellow, Blue} combination is $$2\cdot3\cdot4 = 24.$$

Ways to select {Green, Yellow, Blue} combination is $$2\cdot3\cdot4 = 24.$$

Hence the total number of ways to select 3 distinctly-colored balls $$= 12+16+24+24 = 76.$$

Another example is when $$n = 2,$$ the answer will be $$4$$ Red-Green combinations, $$6$$ Red-Yellow combinations,$$8$$ Red-Blue combinations,$$6$$ Green-Yellow combinations, $$8$$ Green-Blue combinations, $$12$$ Yellow-Blue combinations.

Total $$= 4+6+8+6+8+12 = 44.$$

Last example when $$n = 4,$$ the answer is $$2\cdot2\cdot3\cdot4 = 48.$$

Suppose the $i$th colour has $b_i$ distinguishable balls and there are $t$ different colours.
The second is to consider the coefficient of $x^n$ in the expansion of $$(1+b_1x)(1+b_2x)\cdots(1+b_ix)\cdots.$$
The third is with the recurrence $f(i,c)=f(i,c-1)+b_if(i-1,c-1)$ when $i \le c$ and $f(i,c)=0$ when $i \gt c$, starting with $f(0,c)=1$. You want $f(n,t)$.