Subjects combinatorics

Marbles Distribution 599Bf5

Step-by-step solutions with LaTeX - clean, fast, and student-friendly.

Use the AI math solver

1. **Stating the problem:** Andi has 15 identical marbles and wants to put them into 3 containers such that no container is empty. We need to find the number of ways to do this. 2. **Formula used:** This is a problem of distributing identical items into distinct containers with no container empty. The formula for the number of ways to put $n$ identical items into $k$ distinct containers with no container empty is: $$\binom{n-1}{k-1}$$ 3. **Applying the formula:** Here, $n=15$ and $k=3$, so the number of ways is: $$\binom{15-1}{3-1} = \binom{14}{2}$$ 4. **Calculating $\binom{14}{2}$:** $$\binom{14}{2} = \frac{14 \times 13}{2 \times 1}$$ 5. **Simplifying:** $$= \frac{14 \times 13}{\cancel{2} \times 1} = 7 \times 13 = 91$$ 6. **Answer:** The number of ways Andi can put 15 identical marbles into 3 containers with no container empty is **91**.