In a group of 3 girls and 4 boys, there are two boys B1 and B2. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but B1 and B2 are not adjacent to each other, is:
Get the complete, step-by-step math solution for: "In a group of 3 girls and 4 boys, there are two boys B_1 and B_2. The number of ways, in which these girls and boys can stand in a queue such that all...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Arrange the groups of girls and boys
First, consider the group of 3 girls as a single unit (G) and the group of 4 boys as another single unit (B). These two units can be arranged in a queue in 2! ways. This means either the girls' block comes first, or the boys' block comes first.
Step 2: Arrange the girls within their group
Within the group of 3 girls, the girls can arrange themselves in 3! ways. This is because there are 3 distinct girls, and their internal arrangement matters.
Step 3: Calculate total arrangements for boys without restrictions
Within the group of 4 boys, the boys can arrange themselves in 4! ways if there were no restrictions. This is the total number of permutations for the 4 boys.
Step 4: Calculate arrangements where B1 and B2 are adjacent
To find the number of arrangements where B1 and B2 are adjacent, treat them as a single unit. Now we have 3 units to arrange (the B1B2 pair, B3, and B4), which can be done in 3! ways. Additionally, B1 and B2 can swap positions within their unit in 2! ways.
Step 5: Calculate arrangements where B1 and B2 are NOT adjacent
The number of ways the boys can stand together such that B1 and B2 are not adjacent is the total number of ways to arrange the 4 boys minus the number of ways where B1 and B2 are adjacent.
Step 6: Calculate the total number of ways
Multiply the number of ways to arrange the girl and boy blocks, the number of ways to arrange girls within their block, and the number of ways to arrange boys within their block such that B1 and B2 are not adjacent.