Let m denotes the number of ways in which 5 boys and 5 girls can be arranged in a line alternately and n denotes the number of ways in which 5 boys and 5 girls can be arranged in a circle so that no two boys are together. If m=k n , then the value of k is

Step-by-step solution

There are two alternating gender patterns in a linear arrangement: BGBGBGBGBG and GBGBGBGBGB . Each pattern has 5! arrangements for boys and 5! for girls, contributing 5! 5! ways each. Thus, m = 2 (5! 5!) = 2 (120 120) = 28800 . For circular arrangements with no adjacent boys, first arrange the girls in (5-1)! = 4! = 24 ways around the circle. This creates 5 distinct gaps between girls, where the 5 boys can be arranged in 5! = 120 ways. Hence, n = 4! 5! = 24 120 = 2880 . Given m = k n , we compute k = m n = 28800 2