For today, we will look at the following problem:
Question:
Suppose you have $10$ identical balls and $5$ numbered boxes. How many ways are there to put the balls into the boxes if there are no restrictions?
Formal Solution:
Let the $10$ balls represent the ``stars'' and let the $5-1$ ``bars'' divide the stars into $5$ groups, with each group indicating the number of balls in each box.
Therefore, the number of ways to arrange the stars and bars is equal to $\displaystyle \binom{10+5-1}{10} = 1001.$
This question can be misinterpreted in many ways. The words "no restrictions" could be extremely vague and misinterpreted most commonly as follows: students could potentially interpret the balls as distinguishable in numbered boxes. They take one ball, ask themselves "How many options can I place this ball?", and conclude that for every ball chosen, there are $5$ ways to place them in the distinguishable bins. Therefore, a common answer is $5^{10} = 9765625$ ways. This is incorrect because the balls are in fact indistinguishable, so Ball $A$ in Bin $1$ and Ball $B$ in Bin $2$ is considered the same as Ball $B$ in Bin $1$ and Ball $A$ in Bin $2$.
Let me know your approach to this problem. How did you attempt this problem? Let us know below!