Question

How many ways are there to distribute six indistinguishable balls into nine distinguishable bins?

Short answer

There are 3003 ways to distribute six indistinguishable balls into nine distinguishable bins

Step-by-step solution

Use the formula for integer

Formula for integer

\(C\left( {n + r - 1,r} \right) = \frac{{(n + r - 1)!}}{{n!r!}}\)

n is number of bins

r is indistinguishable balls

Step 2: Solution of number ways 

Let’s, applied n and r values of inequality in integer formula

As the balls are identical the order is not important in which they are distributed.

Here,

\(\begin{array}{l}C\left( {n + r - 1,r} \right) = \frac{{(n + r - 1)!}}{{n!r!}}\\n = 9\\r = 6\\C\left( {n + r - 1,r} \right) = C\left( {9 + 6 - 1,6} \right)\\ = C\left( {14,6} \right)\\ = \frac{{14!}}{{9!6!}}\\ = 3003ways\end{array}\)