An S5 math probid questions

(a)

Ｓ　Ｓ　Ｓ　Ｓ　Ｔ　Ｔ　Ｓ　Ｓ　Ｓ　Ｓ
teachers are in middle, 2P2 ways
A-H are in the rest, 8P8 ways
thus the required no. of arrangements = 2P2 x 8P8 = 80640

(b)

＿Ｔ＿Ｔ＿Ｃ＿Ｄ＿Ｅ＿Ｆ＿Ｇ＿Ｈ＿
arrange teachers and C,D,E,F,G,H, 8P8 ways
A and B in the spaces ”＿”, 9P2 ways
thus the required no. of arrangements = 8P8 x 9P2 = 2903040

(c)

case(i)　＿Ｓ＿Ｓ＿Ｓ＿ＴＴ＿Ｓ＿Ｓ＿Ｓ＿
teachers are in middle, 2P2 ways
arrange C,D,E,F,G,H, 6P6 ways
A in left four spaces and B in right four spaces, 4P1 x 4P1 ways

case(ii)　＿Ｓ＿Ｓ＿Ｓ＿ＴＴ＿Ｓ＿Ｓ＿Ｓ＿
teachers are in middle, 2P2 ways
arrange C,D,E,F,G,H, 6P6 ways
A in right four spaces and B in left four spaces, 4P1 x 4P1 ways

case(iii)　＿Ｓ＿Ｓ＿ＴＴＳＳＳＳ
teachers are in middle, 2P2 ways
arrange C,D,E,F,G,H, 6P6 ways
A and B in the spaces, 3P2 ways

case(iv)　ＳＳＳＳＴＴ＿Ｓ＿Ｓ＿
teachers are in middle, 2P2 ways
arrange C,D,E,F,G,H, 6P6 ways
A and B in the spaces, 3P2 ways

thus the required no. of arrangements
= (2P2 x 6P6 x 4P1 x 4P1) x 2 + (2P2 x 6P6 x 3P2) x 2
= 63360