回復 42# koeagle 的帖子
填充第 1 題
將12個大小寫的英文字母\(A,B,C,D,E,F,a,b,c,d,e,f\)打亂，兩兩任意配成6對，求大小寫同義(如：\(Aa\)為同義配對，\(AB\)、\(Ab\)不是同義配對)至少2對的方法數。
[解答]
反面作法(取捨原理)
任意分組-全不同義-恰1組同義\(=10395-6040-3264=1091\)
任意分組:\(\Pi^{6}_{k=1}{C^{2k}_{2}}/6!=10395\)
全不同義:\(\displaystyle \Pi^{6}_{k=1}{C^{2k}_{2}}/6!-C^{6}_{1}\Pi^{5}_{k=1}{C^{2k}_{2}}/5!+C^{6}_{2}\Pi^{4}_{k=1}{C^{2k}_{2}}/4!-C^{6}_{3}\Pi^{3}_{k=1}{C^{2k}_{2}}/3!+C^{6}_{4}\Pi^{2}_{k=1}{C^{2k}_{2}}/2!-C^{6}_{5}\Pi^{1}_{k=1}{C^{2k}_{2}}/1!+C^{6}_{6}=6040\)
恰1組同義:\(C^{6}_{1}(\Pi^{5}_{k=1}{C^{2k}_{2}}/5!-C^{5}_{1}\Pi^{4}_{k=1}{C^{2k}_{2}}/4!+C^{5}_{2}\Pi^{3}_{k=1}{C^{2k}_{2}}/3!-C^{5}_{3}\Pi^{2}_{k=1}{C^{2k}_{2}}/2!+C^{5}_{4}\Pi^{1}_{k=1}{C^{2k}_{2}}/1!-C^{5}_{5})=6\times 544=3264\)
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