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If 2 boys and 2 girls are to be arranged in a row so that the girls are not next to each other, how many possible arrangements are there?
- a)3
- b)6
- c)12
- d)24
Correct answer is option 'C'. Can you explain this answer?
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Problem Analysis:
We need to arrange 2 boys and 2 girls in a row such that the girls are not next to each other. Let's consider the possible arrangements step by step.
Step 1: Consider the arrangement without any restriction.
We have 4 people to arrange in a row, which can be done in 4! = 4 * 3 * 2 * 1 = 24 ways.
Step 2: Consider the arrangements where the girls are next to each other.
If the girls are next to each other, we can treat them as a single entity.
So, now we have 3 entities to arrange - the entity of the 2 girls and the 2 boys.
This can be done in 3! = 3 * 2 * 1 = 6 ways.
Step 3: Subtract the arrangements where the girls are next to each other from the total arrangements.
The number of possible arrangements where the girls are not next to each other is:
Total arrangements - Arrangements where girls are next to each other = 24 - 6 = 18.
Step 4: Consider the cases where the girls are not next to each other.
Let's consider the 6 arrangements where the girls are next to each other:
1) GG (girls together), BB (boys together)
2) BB, GG
3) GB (girl 1 followed by girl 2), BG (boy 1 followed by boy 2)
4) BG, GB
5) GB, BG
6) BG, GB
Step 5: Count the number of arrangements where the girls are not next to each other.
Now, let's count the arrangements where the girls are not next to each other:
1) GB, BG
2) GB, BG
3) BG, GB
4) BG, GB
5) GB, BG
6) GB, BG
7) BG, GB
8) GB, BG
9) BG, GB
10) GB, BG
11) BG, GB
12) GB, BG
Hence, there are 12 possible arrangements where the girls are not next to each other.
Therefore, the correct answer is option 'C' - 12.
We need to arrange 2 boys and 2 girls in a row such that the girls are not next to each other. Let's consider the possible arrangements step by step.
Step 1: Consider the arrangement without any restriction.
We have 4 people to arrange in a row, which can be done in 4! = 4 * 3 * 2 * 1 = 24 ways.
Step 2: Consider the arrangements where the girls are next to each other.
If the girls are next to each other, we can treat them as a single entity.
So, now we have 3 entities to arrange - the entity of the 2 girls and the 2 boys.
This can be done in 3! = 3 * 2 * 1 = 6 ways.
Step 3: Subtract the arrangements where the girls are next to each other from the total arrangements.
The number of possible arrangements where the girls are not next to each other is:
Total arrangements - Arrangements where girls are next to each other = 24 - 6 = 18.
Step 4: Consider the cases where the girls are not next to each other.
Let's consider the 6 arrangements where the girls are next to each other:
1) GG (girls together), BB (boys together)
2) BB, GG
3) GB (girl 1 followed by girl 2), BG (boy 1 followed by boy 2)
4) BG, GB
5) GB, BG
6) BG, GB
Step 5: Count the number of arrangements where the girls are not next to each other.
Now, let's count the arrangements where the girls are not next to each other:
1) GB, BG
2) GB, BG
3) BG, GB
4) BG, GB
5) GB, BG
6) GB, BG
7) BG, GB
8) GB, BG
9) BG, GB
10) GB, BG
11) BG, GB
12) GB, BG
Hence, there are 12 possible arrangements where the girls are not next to each other.
Therefore, the correct answer is option 'C' - 12.
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If 2 boys and 2 girls are to be arranged in a row so that the girls are not next to each other, how many possible arrangements are there?a)3b)6c)12d)24Correct answer is option 'C'. Can you explain this answer?
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