Question:
Number of ways 3 boys and 4 girls can be arranged such that there is one girl between any 2 boys and one boy between any 2 girls.
Number of ways 3 boys and 4 girls can be arranged such that there is one girl between any 2 boys and one boy between any 2 girls.
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In such arrangement problems, first fix the positions of one group and then place the other group in the available positions.
Updated On: Apr 18, 2026
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Solution and Explanation
Step 1: Arrangement of boys and girls.
Since there must be one girl between any two boys and one boy between any two girls, we first arrange the boys and then place the girls in the available positions. The boys can be arranged in \( 3! \) ways. After placing the boys, there are 4 positions between the boys to place the girls.
Step 2: Arrangement of girls.
The girls can be arranged in \( 4! \) ways in the 4 available positions.
Step 3: Total number of arrangements.
The total number of ways to arrange the boys and girls is the product of the number of ways to arrange the boys and the number of ways to arrange the girls: \[ 3! \times 4! \]
Since there must be one girl between any two boys and one boy between any two girls, we first arrange the boys and then place the girls in the available positions. The boys can be arranged in \( 3! \) ways. After placing the boys, there are 4 positions between the boys to place the girls.
Step 2: Arrangement of girls.
The girls can be arranged in \( 4! \) ways in the 4 available positions.
Step 3: Total number of arrangements.
The total number of ways to arrange the boys and girls is the product of the number of ways to arrange the boys and the number of ways to arrange the girls: \[ 3! \times 4! \]
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