Number of Ways of Arranging 5 Identical Balls to 3 Different Bags

There are several ways to approach this problem. One way is to use a combination of stars and bars method.

Step 1: Visualize the Problem
We have 5 identical balls and 3 different bags to place them in. We need to find the number of ways we can arrange these balls in the bags.

Step 2: Understanding the Stars and Bars Method
The stars and bars method is a combinatorial technique used to count the number of ways to distribute identical objects into distinct groups. In our case, the stars represent the balls and the bars represent the boundaries between the bags.

Step 3: Applying the Stars and Bars Method
To solve this problem, we can use the stars and bars method with a slight modification. We will use 2 bars to divide the 5 stars (balls) into 3 groups (bags).

Step 4: Arranging the Stars and Bars
We can visualize the arrangement as follows:

* | * * * | *

In this arrangement, the first bag contains 1 ball, the second bag contains 3 balls, and the third bag contains 1 ball. We can also represent this arrangement using numbers:

1 3 1

Step 5: Counting the Arrangements
We need to find the number of ways we can arrange the stars and bars. In this case, we have 2 bars and 5 stars, so we can use the formula:

C(n+k-1, k-1)

where n is the number of stars and k is the number of bars. Plugging in the values, we get:

C(5+2-1, 2-1) = C(6, 1) = 6

Therefore, there are 6 different ways to arrange the 5 identical balls into 3 different bags.

Summary
Using the stars and bars method, we determined that there are 6 different ways to arrange 5 identical balls into 3 different bags. This method allows us to count the number of ways to distribute identical objects into distinct groups, and can be applied to various combinatorial problems.