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In a class of 10 students there are 3 girls. The number of ways they can be arranged in a row, so that no two girls are consecutive is k. 8!, where k =
- a)42
- b)12
- c)24
- d)36
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
In a class of 10 students there are 3 girls. The number of ways they c...
Given:
- Total number of students in the class = 10
- Number of girls in the class = 3
To find:
- The number of ways the girls can be arranged in a row, such that no two girls are consecutive
Approach:
- We can solve this problem by using the principle of permutations.
- We will first find the total number of ways the students can be arranged, and then subtract the number of arrangements where two or more girls are consecutive.
Calculation:
1. Total number of ways to arrange the students:
- Since there are 10 students in total, we can arrange them in 10! ways (10 factorial).
2. Number of ways to arrange the students with two or more girls consecutive:
- We can treat the group of girls as a single entity (G) and arrange the remaining students (7 students) and the group of girls in 8! ways.
- Within the group of girls, there are 3! ways to arrange the girls themselves.
3. Subtracting the arrangements with two or more girls consecutive from the total number of arrangements:
- Number of ways with two or more girls consecutive = 8! * 3!
- Number of ways without two or more girls consecutive = 10! - (8! * 3!)
4. Simplifying the expression:
- Number of ways without two or more girls consecutive = 10! - (8! * 3!)
- Number of ways without two or more girls consecutive = 8! * (10 * 9 - 3)
- Number of ways without two or more girls consecutive = 8! * (90 - 3)
- Number of ways without two or more girls consecutive = 8! * 87
5. Final answer:
- Given, k = 8!
- k = 8! * 87
- Therefore, k = 8! = 40320
Conclusion:
The number of ways the girls can be arranged in a row, such that no two girls are consecutive, is 40320, which is equal to 8!. Hence, the correct answer is option 'A'.
- Total number of students in the class = 10
- Number of girls in the class = 3
To find:
- The number of ways the girls can be arranged in a row, such that no two girls are consecutive
Approach:
- We can solve this problem by using the principle of permutations.
- We will first find the total number of ways the students can be arranged, and then subtract the number of arrangements where two or more girls are consecutive.
Calculation:
1. Total number of ways to arrange the students:
- Since there are 10 students in total, we can arrange them in 10! ways (10 factorial).
2. Number of ways to arrange the students with two or more girls consecutive:
- We can treat the group of girls as a single entity (G) and arrange the remaining students (7 students) and the group of girls in 8! ways.
- Within the group of girls, there are 3! ways to arrange the girls themselves.
3. Subtracting the arrangements with two or more girls consecutive from the total number of arrangements:
- Number of ways with two or more girls consecutive = 8! * 3!
- Number of ways without two or more girls consecutive = 10! - (8! * 3!)
4. Simplifying the expression:
- Number of ways without two or more girls consecutive = 10! - (8! * 3!)
- Number of ways without two or more girls consecutive = 8! * (10 * 9 - 3)
- Number of ways without two or more girls consecutive = 8! * (90 - 3)
- Number of ways without two or more girls consecutive = 8! * 87
5. Final answer:
- Given, k = 8!
- k = 8! * 87
- Therefore, k = 8! = 40320
Conclusion:
The number of ways the girls can be arranged in a row, such that no two girls are consecutive, is 40320, which is equal to 8!. Hence, the correct answer is option 'A'.
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In a class of 10 students there are 3 girls. The number of ways they c...
7! x (8P3) = k x 8!
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