【数学A】例題1.2.7：重複順列（One More）★★

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題A1.2.7：重複順列（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) 5人でじゃんけんを1回するとき，5人のグー，チョキ，パーの手の出し方は何通りあるか．

(2) 集合$A=\{a,b,c,d,e\}$の部分集合は全部で何個あるか．

(3) 0，1，2，3の4個の数字の中から，重複を許して4個取って1列に並べるとき，4桁の整数は何個できるか．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題A1.2.7の解答（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) 1人の手の出し方はグー，チョキ，パーの3通りずつある．

よって，5人の手の出し方は，

\[
3\times 3\times 3\times 3\times 3=3^5=243(\text{通り})
\]

(2) 集合$A$の部分集合の個数は，$A$の5つの要素$a,b,c,d,e$のそれぞれが，部分集合に属しているか否かの決め方の数だけある．

よって，集合$A$の部分集合の個数は，

\[
2^5=32\text{(個)}
\]

(3) 千の位に並べる数は，1，2，3の3通り

そのそれぞれについて，百の位と十の位と一の位に並べる数は，0，1，2，3のいずれでもよいから，その個数は，$4^3$通り

よって，求める4桁の整数の個数は，

\[
3\times 4^3=192(\text{個})
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題A1.2.7：重複順列（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) 集合$A=\{x,y,z,w\}$の部分集合は全部で何個あるか．

(2) 0，1，2，3，4の5個の数字の中から，重複を許して3個取って1列に並べるとき，3桁の整数は何個できるか．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題A1.2.7の解答（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) 集合$A$の部分集合の個数は，$A$の4つの要素$x,y,z,w$のそれぞれが，部分集合に属しているか否かの決め方の数だけある．

よって，集合$A$の部分集合の個数は，

\[
2^4=16\text{(個)}
\]

(2) 百の位に並べる数は，1，2，3，4の4通り

そのそれぞれについて，十の位と一の位に並べる数は，0，1，2，3，4のいずれでもよいから，その個数は，$5^2$通り

よって，求める3桁の整数の個数は，

\[
4\times 5^2=100(\text{個})
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題A1.2.7の側注：重複順列（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
$\underbrace{3\times 3\times 3\times 3\times 3}_{5\text{個}}$

% 側注2
部分集合は，$A$自身である$A=\{a,b,c,d,e\}$と空集合$\varnothing$を含むことに注意すること．

% 側注3
千の位は0ではないことに注意すること．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題A1.2.7の側注：重複順列（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
部分集合は，$A$自身である$A=\{x,y,z,w\}$と空集合$\varnothing$を含むことに注意すること．

% 側注2
百の位は0ではないことに注意すること．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題A1.2.7の考え方・補足など：重複順列（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 考え方
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) 5人がそれぞれ3通りずつの手を出すことができるので，$3^{5}$通りを考える．

(2) すべての部分集合を書き出して求めようとすると，手間が掛かる．部分集合は，それぞれの要素がその部分集合に属しているか否かで決まるので，(1)と同様に$n^r$を考える．部分集合に属している場合を$\bigcirc$，属していない場合を$\times$で表すと，右の表のようになる．5個の要素がそれぞれ部分集合に含まれるか否かの2通りで決まるので，$2^{5}$通りを考える．

\begin{tabular}{|ccccc|c|}
\hline
a&b&c&d&e&\\
\hline
$\bigcirc$&$\bigcirc$&$\bigcirc$&$\bigcirc$&$\bigcirc$&$\{a,b,c,d,e\}$\\
$\bigcirc$&$\bigcirc$&$\bigcirc$&$\bigcirc$&$\times$&$\{a,b,c,d\}$\\
$\bigcirc$&$\bigcirc$&$\bigcirc$&$\times$&$\bigcirc$&$\{a,b,c,e\}$\\
$\vdots$&$\vdots$&$\vdots$&$\vdots$&$\vdots$&$\vdots$\\
$\times$&$\times$&$\times$&$\times$&$\times$&$\varnothing$\\
\hline
\end{tabular}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% One Point
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

重複を許して並べるときは，重複順列$n^r$を用いる．