Final answer:
There are 1820 ways to distribute 12 identical balls into 5 distinct boxes, calculated using the stars and bars combinatorial method.
Step-by-step explanation:
The question asks about the number of ways to distribute 12 identical balls into 5 distinct boxes. To solve this, we use a combinatorial method known as stars and bars. The problem is equivalent to placing 12 indistinguishable items into 5 distinguishable containers.
In combinatorics, we can imagine this process as placing 12 stars in a row, and using 4 bars to divide them into 5 groups. Each group of stars represents the number of balls in a box. Every arrangement of stars and bars corresponds to a unique way to distribute the balls into the boxes.
The formula for this kind of problem is given by the combination of (n + k - 1) choose (k - 1), where n is the number of items to distribute, and k is the number of containers. In this case, n=12 and k=5. So we compute (12 + 5 - 1) choose (4), which simplifies to 16 choose 4.
The calculation for 16 choose 4 is:
16 choose 4 = 16! / (4! * (16 - 4)!) = 16! / (4! * 12!) = (16 * 15 * 14 * 13) / (4 * 3 * 2 * 1) = 1820
Therefore, there are 1820 ways to distribute 12 identical balls into 5 distinct boxes.