The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is ?

To find the number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty, we can use the concept of stars and bars or balls and urns.

Explanation:

When distributing the balls, we can imagine using 8 stars and 2 bars to represent the three boxes. The bars divide the stars into three groups, representing the balls in each box. The positions of the bars determine the distribution of the balls.

For example, if we have 8 stars and 2 bars arranged like this: *|**|****, it means the first box has 1 ball, the second box has 2 balls, and the third box has 4 balls.

Using the concept of stars and bars:

The number of ways to arrange the 8 stars and 2 bars can be calculated using the formula: (n + k - 1) choose (k - 1), where n represents the number of stars and k represents the number of bars.

In this case, n = 8 and k = 2, so the number of ways is (8 + 2 - 1) choose (2 - 1) = 9 choose 1 = 9.

However, this includes the case where one of the boxes is empty, which is not allowed in the question. So, we need to subtract the number of ways where one box is empty.

Subtracting the cases where one box is empty:

To find the number of ways where one box is empty, we can distribute 8 balls in 2 boxes and then distribute the remaining 0 balls in the third box.

Using the same concept of stars and bars, we have n = 8 and k = 2. So, the number of ways to distribute the 8 balls in 2 boxes is (8 + 2 - 1) choose (2 - 1) = 9 choose 1 = 9.

Therefore, the number of ways where one box is empty is 9.

Calculating the final answer:

To find the number of ways where none of the boxes is empty, we subtract the cases where one box is empty from the total number of ways.

Total ways = 9
Ways where one box is empty = 9

Number of ways where none of the boxes is empty = Total ways - Ways where one box is empty
= 9 - 9
= 0

Therefore, the correct answer is (1) 0.

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