How many ways can 5 identical red balls and 4 identical blue balls be arranged in a row such that no two blue balls are adjacent?

The problem can be solved using the concept of permutations and combinations.

Step 1: Arrange the red balls

First, arrange the 5 identical red balls in a row. Since they are identical, there is only 1 way to do this.

R R R R R

Step 2: Identify the spaces between and at the ends of the red balls

There are 6 spaces where the blue balls can be placed (indicated by the underscores):

_ R _ R _ R _ R _ R _

Step 3: Place the blue balls

We have 4 identical blue balls and 6 spaces. We need to choose 4 spaces out of 6 to place the blue balls. Since the blue balls are identical, this is a combination problem.

The number of ways to choose 4 spaces out of 6 is given by the combination formula C(n, r) = n! / [r!(n-r)!], where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.

So, the number of ways to place the blue balls is C(6, 4) = 6! / [4!(6-4)!] = 15.

Therefore, there are 15 ways to arrange 5 identical red balls and 4 identical blue balls in a row such that no two blue balls are adjacent.

The problem can be solved using the concept of permutations and combinations.

Step 1: Consider the 5 red balls to be arranged in a row. This can be done in only 1 way since they are identical.

Step 2: Now, we need to place the 4 blue balls such that no two blue balls are adjacent. This can be done by placing them in the gaps between the red balls. There are 6 gaps between and around the red balls (before the first red ball, between the first and second red balls, between the second and third red balls, between the third and fourth red balls, between the fourth and fifth red balls, and after the fifth red ball).

Step 3: We need to select 4 out of these 6 gaps to place the blue balls. This can be done in 6C4 ways.

6C4 = 6! / (4!(6-4)!) = 15

So, there are 15 ways to arrange 5 identical red balls and 4 identical blue balls in a row such that no two blue balls are adjacent.