Question

How many ways are there to distribute 12 indistinguishable balls into six distinguishable bins?

Short answer

There are 6188 ways to distribute 12 indistinguishable balls into six distinguishable bins

Step-by-step solution

Step 1: 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,

\(\begin{array}{l}C\left( {n + r - 1,r} \right) = \frac{{(n + r - 1)!}}{{n!r!}}\\n = 6\\r = 12\\C\left( {n + r - 1,r} \right) = C\left( {6 + 12 - 1,12} \right)\\ = C\left( {17,12} \right)\\ = \frac{{17!}}{{6!12!}}\\ = 6188ways\end{array}\)