In how many ways can 5 identical coins be placed in 3 different pockets?
Consider the following figure (with two bars):

$$
\left|\begin{array}{lllllll}
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right|
$$

Replace each circle with either a piece or a bar so that there are a total of 2 bars and 5 pieces. The pieces between the first two bars will be contained in the first pocket, the pieces between the second and third bars will be contained in the second pocket, and finally, the pieces between the third and fourth bars will be contained in the third pocket.

Thus, placing 5 identical pieces into 3 different pockets is equivalent to choosing the positions of the 2 bars among 7 possible positions. The answer is therefore $\binom{7}{2}=21$.

More generally, proceeding in the same way, we see that there are $\binom{a+b-1}{b-1}=\binom{a+b-1}{a}$ ways to place $a$ identical pieces into $b$ different pockets.

## 4 TD of Basic Strategies

## - Exercises -