This is a classic combinatorics problem where you need to find the number of ways to distribute $10$ identical ping pong balls into $6$ labeled cups. This is a case of the stars and bars problem, which deals with distributing identical objects (ping pong balls) into distinct groups (cups).

The formula for this is:

$\text{Number of ways} = \binom{n + k - 1}{k - 1}$

where:

• $n$ is the number of identical objects (ping pong balls) = 10,
• $k$ is the number of groups (cups) = 6.

Using the formula, we calculate:

$\binom{10 + 6 - 1}{6 - 1} = \binom{15}{5}$

The expression $\binom{15}{5}$ is equivalent to $\binom{15}{10}$, and this corresponds to C(15,9) in the provided options.

Thus, the correct answer is:

e) C(15,9)

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Would you like a detailed explanation of how the stars and bars method works? Or do you have any other questions on this topic?

Here are 5 related questions:

1. How do you calculate $\binom{n}{k}$ using the formula?
2. What happens if the cups were also identical (not labeled)?
3. What is the difference between permutations and combinations in combinatorics?
4. Can you solve a similar problem where there are multiple types of balls?
5. How would you solve the problem if the number of balls in each cup had a specific constraint (e.g., at least 1 ball in each cup)?

Tip: When using the stars and bars formula, make sure to subtract 1 from the number of cups in the denominator, as you're finding the number of ways to partition the objects between groups.