問題の解答
検索用コード（LaTeX）
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% 例題A1.2.6：条件付きの円順列（One More）★★
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両親と息子2人，娘2人の合計6人が円卓に座るとき，次の問いに答えよ．
(1) 両親が正面に向かい合う座り方は何通りあるか．
(2) 男性と女性が交互になる座り方は何通りあるか．
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% 例題A1.2.6の解答（One More）★★
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(1) 父の席を固定すると，母の席は正面に1通りとなる．残りの4人の子供の座り方は，4箇所に並べる順列であるから，$4!$通り
よって，両親が正面に向かい合う座り方は，
\[
4!=4\cdot 3\cdot 2\cdot 1=24(\text{通り})
\]
(2) 男性3人が円形に座る座り方は，$(3-1)!$通り
そのそれぞれについて，間に入る残りの3人の女性の座り方は，3箇所に並べる順列であるから，$3!$通り
よって，男性と女性が交互になる座り方は，
\[
(3-1)!\times 3!=12(\text{通り})
\]
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% 問題A1.2.6：条件付きの円順列（One More）★★
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両親と息子3人，娘3人の合計8人が円卓に座るとき，次の問いに答えよ．
(1) 両親が正面に向かい合う座り方は何通りあるか．
(2) 男性と女性が交互になる座り方は何通りあるか．
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% 問題A1.2.6の解答（One More）★★
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(1) 父の席を固定すると，母の席は正面に1通りとなる．残りの6人の座り方は，{6箇所に並べる順列}であるから，$6!$通り
よって，両親が正面に向かい合う座り方は，
\[
6!=6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1=720(\text{通り})
\]
(2) 男性4人が円形に座る座り方は，$(4-1)!$通り
そのそれぞれについて，間に入る残りの4人の女性の座り方は，{4箇所に並べる順列}であるから，$4!$通り
よって，男性と女性が交互になる座り方は，
\[
(4-1)!\times 4!=144(\text{通り})
\]
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