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CAT Exam  >  CAT Questions  >  Boxes numbered 1,2,3,4 and 5 are kept in a ro... Start Learning for Free
Boxes numbered 1,2,3,4 and 5 are kept in a row and they are filled with either a red or a blue ball, such that two adjacent boxes can be filled with blue balls. Then how many different arrangements are possible, given that all balls or given colour are exactly identical in all respects?
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Most Upvoted Answer
Boxes numbered 1,2,3,4 and 5 are kept in a row and they are filled wit...
**Problem Analysis:**
We are given 5 boxes arranged in a row. Each box can be filled with either a red or a blue ball. The condition is that two adjacent boxes can be filled with blue balls. We need to find the number of different arrangements possible.

**Solution:**
To solve this problem, we can use a combination of counting and reasoning.

**Step 1: Counting the number of arrangements with no restrictions**
Let's first consider the case where there are no restrictions on the arrangement of the balls in the boxes. In this case, each box can be filled with either a red or a blue ball. So, for each box, there are two possibilities - red or blue. Since there are 5 boxes, the total number of arrangements with no restrictions is 2^5 = 32.

**Step 2: Counting the number of arrangements with the given restriction**
Now, let's consider the case where the restriction is that two adjacent boxes can be filled with blue balls. We need to count the number of arrangements that satisfy this restriction.

**Case 1: Two blue balls at the beginning**
In this case, the first two boxes are filled with blue balls. The remaining three boxes can be filled with either red or blue balls. So, for the remaining three boxes, there are 2^3 = 8 possibilities. Therefore, the total number of arrangements in this case is 8.

**Case 2: Two blue balls at the end**
In this case, the last two boxes are filled with blue balls. The remaining three boxes can be filled with either red or blue balls. So, for the remaining three boxes, there are 2^3 = 8 possibilities. Therefore, the total number of arrangements in this case is 8.

**Case 3: Two blue balls in the middle**
In this case, the middle three boxes are filled with blue balls. The first and the last boxes can be filled with either red or blue balls. So, for the first and the last boxes, there are 2 possibilities each. Therefore, the total number of arrangements in this case is 2 * 2 = 4.

**Total number of arrangements with the given restriction**
To find the total number of arrangements with the given restriction, we add up the number of arrangements in each case.
Total number of arrangements = number of arrangements in Case 1 + number of arrangements in Case 2 + number of arrangements in Case 3
Total number of arrangements = 8 + 8 + 4 = 20

**Conclusion:**
Therefore, the total number of different arrangements possible, given that two adjacent boxes can be filled with blue balls, is 20.
Community Answer
Boxes numbered 1,2,3,4 and 5 are kept in a row and they are filled wit...
There are many number of possibilities, as it is not specifically mentioned that only 2 adjacent boxes mudt contain blue balls.
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Boxes numbered 1,2,3,4 and 5 are kept in a row and they are filled with either a red or a blue ball, such that two adjacent boxes can be filled with blue balls. Then how many different arrangements are possible, given that all balls or given colour are exactly identical in all respects?
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