How many ways can 5 boys and 5 girls be seated at a round table if: a. no restrictions are imposed b. the girls and boys are to occupy alternate seats c. 3 particulars girls must sit together d. 3 particulars girls must not sit together e. all the girls must sit together Solution: a. The number of arrangements of 10 persons to be seated at a round table is (n-1)! = (10-1)! = 9! = 362,880 b. ________________________________ _ _ c. Consider the 3 particular girls as 1 group or unit. Since there are 2 girls and 5 boys to be seated, there area total of:5 seats for boys + 2 seats for girls + 1 group seat = 8seats (n) to be arranged in a circle . Thus, (n-1)! = (8-1)! = 7! = 5, 040 Moreover, the 3 girls can be arranged within its group in (3) (2) (1) = 6ways. By FPC, the required number of ways the 10 persos can be seated is 5, 040 x 6 = 30,240 d. . ___________________________________

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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How many ways can 5 boys and 5 girls be seated at a round table if: a. no restrictions are imposed b. the girls and boys are to occupy alternate seats c. 3 particulars girls must sit together d. 3 particulars girls must not sit together e. all the girls must sit together Solution: a. The number of arrangements of 10 persons to be seated at a round table is (n-1)! = (10-1)! = 9! = 362,880 b. __________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ ____________________ c. Consider the 3 particular girls as 1 group or unit. Since there are 2 girls and 5 boys to be seated, there area total of:5 seats for boys + 2 seats for girls + 1 group seat = 8seats (n) to be arranged in a circle . Thus, (n-1)! = (8-1)! = 7! = 5, 040 Moreover, the 3 girls can be arranged within its group in (3) (2) (1) = 6ways. By FPC, the required number of ways the 10 persos can be seated is 5, 040 x 6 = 30,240 d. . ____________________________________________________________________ ____________________________________________________________________ _____________________________________________________________________ e. Treat the 5 girls as 1 group 5 seats for boys + 1 group seats for girls = 6 sets to be arranged in a circle. (n-1)! = (6-1)! = 5! = 120 Moreover , the 5 girls can be arranged within its group in 5! ways . By FPC, the required number of ways the 10 persons can be seated is (120) (5!) = (120) (120) = 14, 400

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