In how many ways can 8 identical balls be distributed among 3 distinct boxes such that no box is empty?
45
60
75
81
To solve this problem, we can use a combinatorial approach known as "stars and bars" or "balls and urns."We have 8 identical balls to distribute among 3 distinct boxes, and we want to ensure that no box is empty. This means that each box must have at least one ball.
Let's represent the balls using stars (?) and the divisions between the boxes using bars (|). We have 8 balls, so we'll have 8 stars (????????) and we need to place 2 bars to divide them into 3 boxes.
For example, one possible arrangement could be: ??|???|??
The number of ways to arrange these stars and bars will give us the number of ways to distribute the balls among the boxes. To calculate the total number of arrangements, we need to determine the positions of the 2 bars among the 8+2 =10 total positions (8 stars and 2 bars). We can choose 2 positions out of 10 to place the bars, which can be calculated as
Therefore, there are 45 ways to distribute the 8 identical balls among the 3 distinct boxes such that no box is empty.
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