次の値を求めよう。
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\vn{10} \def\vr{3} \def\abs{9 \cdot 8} \def\abb{2 \cdot 1} \def\yakubun{10 \cdot 3 \cdot 4} \def\kotae{120} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} & \colMM{orange}{ \vn から}\colMM{green}{\vr 個\rightarrow}\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= \dfrac{\colBX{bisque}{\vn}\cdot\abs}{\colBX{palegreen}{$\vr$}\cdot\abb}\\ & \colMM{green}{ \vr から1まで}\\ &= \yakubun\\ \\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\vn{5} \def\vr{2} \def\abs{4} \def\abb{1} \def\yakubun{5 \cdot 2} \def\kotae{10} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} & \colMM{orange}{ \vn から}\colMM{green}{\vr 個\rightarrow}\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= \dfrac{\colBX{bisque}{\vn}\cdot\abs}{\colBX{palegreen}{$\vr$}\cdot\abb}\\ & \colMM{green}{ \vr から1まで}\\ &= \yakubun\\ \\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\vn{8} \def\vr{4} \def\abs{7 \cdot 6 \cdot 5} \def\abb{3 \cdot 2 \cdot 1} \def\yakubun{2 \cdot 7 \cdot 5} \def\kotae{70} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} & \colMM{orange}{ \vn から}\colMM{green}{\vr 個\rightarrow}\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= \dfrac{\colBX{bisque}{\vn}\cdot\abs}{\colBX{palegreen}{$\vr$}\cdot\abb}\\ & \colMM{green}{ \vr から1まで}\\ &= \yakubun\\ \\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\vn{7} \def\vr{1} \def\kotae{7} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} & \colMM{orange}{ \vn から}\colMM{green}{\vr 個\rightarrow}\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= \dfrac{\colBX{bisque}{\vn}}{\colBX{palegreen}{$\vr$}}\\ & \colMM{green}{ \vr から1まで}\\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】定義通りに解く
\def\vn{6} \def\vr{6} \def\abs{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \def\abb{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \def\yakubun{2 \cdot 7 \cdot 5} \def\kotae{1} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} & \colMM{orange}{ \vn から}\colMM{green}{\vr 個\rightarrow}\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= \dfrac{\colBX{bisque}{\vn}\cdot\abs}{\colBX{palegreen}{$\vr$}\cdot\abb}\\ & \colMM{green}{ \vr から1まで}\\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
【解答】公式として覚えてしまうのもアリ!
\def\vn{6} \def\vr{6} \def\kotae{1} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{red}{そろったら\Darr}\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} = \colBX{mistyrose}{$\kotae$} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
【解答】変換かけて小さく!
\def\vn{6} \def\vr{6} \def\vs{0} \def\kotae{1} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{green}{\Darr 足して\ \vn\ \Darr \ }\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} = {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vs$}} &= \kotae\\ \colMM{red}{ {}_{n}{\rm C}_{0}} & \colMM{red}{=1}\\ \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
次の値を求めよう。
↓この問題へのリンクはこちら(右クリックで保存)
【解答】変換かけて楽する!
\def\vn{11} \def\vr{9} \def\vs{2} \def\abs{10} \def\abb{1} \def\bbs{11 \cdot 10} \def\bbb{2 \cdot 1} \def\yakubun{11 \cdot 5} \def\kotae{55} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{green}{\Darr\ } & \colMM{green}{足して\ \vn\ \Darr \ }\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vs$}}\\ \\ & \colMM{orange}{ \vn から}\colMM{green}{\vs 個\rightarrow}\\ &= \dfrac{\colBX{bisque}{\vn}\cdot\abs}{\colBX{palegreen}{$\vs$}\cdot\abb}\\ & \colMM{green}{ \vs から1まで}\\ \\ &= \yakubun\\ \\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
【解答】定義通りに解く
\def\vn{11} \def\vr{9} \def\vs{2} \def\abs{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3} \def\abb{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \def\bbs{11 \cdot 10} \def\bbb{2 \cdot 1} \def\yakubun{11 \cdot 5} \def\kotae{55} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} & \colMM{orange}{ \vn から}\colMM{green}{\vr 個\rightarrow}\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= \dfrac{\colBX{bisque}{\vn}\cdot\abs}{\colBX{palegreen}{$\vr$}\cdot\abb}\\ & \colMM{green}{ \vr から1まで}\\ \\ &= \dfrac{\bbs}{\bbb} \colMM{lightgray}{ ={}_{\vn}{\rm C}_{\vs} だね。}\\ \\ &= \yakubun\\ \\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】変換かけて楽する!
\def\vn{8} \def\vr{6} \def\vs{2} \def\abs{7} \def\abb{1} \def\yakubun{4 \cdot 7} \def\kotae{28} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{green}{\Darr\ } & \colMM{green}{足して\ \vn\ \Darr \ }\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vs$}}\\ \\ & \colMM{orange}{ \vn から}\colMM{green}{\vs 個\rightarrow}\\ &= \dfrac{\colBX{bisque}{\vn}\cdot\abs}{\colBX{palegreen}{$\vs$}\cdot\abb}\\ & \colMM{green}{ \vs から1まで}\\ \\ &= \yakubun\\ \\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
【解答】定義通りに解く
\def\vn{8} \def\vr{6} \def\vs{2} \def\abs{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3} \def\abb{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \def\bbs{8 \cdot 7} \def\bbb{2 \cdot 1} \def\yakubun{4 \cdot 7} \def\kotae{28} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} & \colMM{orange}{ \vn から}\colMM{green}{\vr 個\rightarrow}\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= \dfrac{\colBX{bisque}{\vn}\cdot\abs}{\colBX{palegreen}{$\vr$}\cdot\abb}\\ & \colMM{green}{ \vr から1まで}\\ \\ &= \dfrac{\bbs}{\bbb} \colMM{lightgray}{ ={}_{\vn}{\rm C}_{\vs} だね。}\\ \\ &= \yakubun\\ \\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】変換かけて楽する!
\def\vn{10} \def\vr{7} \def\vs{3} \def\abs{9 \cdot 8} \def\abb{2 \cdot 1} \def\yakubun{10 \cdot 3 \cdot 4} \def\kotae{120} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{green}{\Darr\ } & \colMM{green}{足して\ \vn\ \Darr \ }\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vs$}}\\ \\ & \colMM{orange}{ \vn から}\colMM{green}{\vs 個\rightarrow}\\ &= \dfrac{\colBX{bisque}{\vn}\cdot\abs}{\colBX{palegreen}{$\vs$}\cdot\abb}\\ & \colMM{green}{ \vs から1まで}\\ \\ &= \yakubun\\ \\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
【解答】定義通りに解く
\def\vn{10} \def\vr{7} \def\vs{3} \def\abs{9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4} \def\abb{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \def\bbs{10 \cdot 9 \cdot 8} \def\bbb{3 \cdot 2 \cdot 1} \def\yakubun{10 \cdot 3 \cdot 4} \def\kotae{120} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} & \colMM{orange}{ \vn から}\colMM{green}{\vr 個\rightarrow}\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= \dfrac{\colBX{bisque}{\vn}\cdot\abs}{\colBX{palegreen}{$\vr$}\cdot\abb}\\ & \colMM{green}{ \vr から1まで}\\ \\ &= \dfrac{\bbs}{\bbb} \colMM{lightgray}{ ={}_{\vn}{\rm C}_{\vs} だね。}\\ \\ &= \yakubun\\ \\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】変換かけて楽する!
\def\vn{20} \def\vr{18} \def\vs{2} \def\abs{19} \def\abb{1} \def\yakubun{10 \cdot 19} \def\kotae{190} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{green}{\Darr\ } & \colMM{green}{足して\ \vn\ \Darr \ }\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vs$}}\\ \\ & \colMM{orange}{ \vn から}\colMM{green}{\vs 個\rightarrow}\\ &= \dfrac{\colBX{bisque}{\vn}\cdot\abs}{\colBX{palegreen}{$\vs$}\cdot\abb}\\ & \colMM{green}{ \vs から1まで}\\ \\ &= \yakubun\\ \\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
【解答】定義通りに解く
\def\vn{20} \def\vr{18} \def\vs{2} \def\abs{19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3} \def\abb{17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \def\bbs{20 \cdot 19} \def\bbb{2 \cdot 1} \def\yakubun{10 \cdot 19} \def\kotae{190} \newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} & \colMM{orange}{ \vn から}\colMM{green}{\vr 個\rightarrow}\\ {}_{\colBX{bisque}{$\scriptsize \vn$}}{\rm C}_{\colBX{palegreen}{$\scriptsize\vr$}} &= \dfrac{\colBX{bisque}{\vn}\cdot\abs}{\colBX{palegreen}{$\vr$}\cdot\abb}\\ & \colMM{green}{ \vr から1まで}\\ \\ &= \dfrac{\bbs}{\bbb} \colMM{lightgray}{ ={}_{\vn}{\rm C}_{\vs} だね。}\\ \\ &= \yakubun\\ \\ &= \kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
次の各問いに答えよ。
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & \colBX{bisque}{$8$}\ 個の頂点から\\ & 異なる\ \colBX{palegreen}{$3$}\ つの頂点を選べば\\ & 三角形が1個できるから,\\ \\ & 求める三角形の個数は\\ & {}_{\colBX{bisque}{$\scriptsize 8$}}{\rm C}_{\colBX{palegreen}{$\scriptsize 3$}} = \dfrac{\colBX{bisque}{$8$}\cdot 7 \cdot 6}{\colBX{palegreen}{$3$}\cdot 2 \cdot 1} = 8 \cdot 7 = 56(個) \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & \colBX{bisque}{$8$}\ 個の頂点から\\ & 異なる\ \colBX{palegreen}{$4$}\ つの頂点を選べば\\ & 四角形が1個できるから,\\ \\ & 求める四角形の個数は\\ & {}_{\colBX{bisque}{$\scriptsize 8$}}{\rm C}_{\colBX{palegreen}{$\scriptsize 4$}} = \dfrac{\colBX{bisque}{$8$}\cdot 7 \cdot 6 \cdot 5}{\colBX{palegreen}{$4$}\cdot 3 \cdot 2 \cdot 1} = 2 \cdot 7 \cdot 5 = 70(個) \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan