数学 - JavaScript - 「場合の数」 を数える - 順列・組合せ – 順列 - 組合せの公式

数学読本〈4〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

数学読本〈4〉数列の極限，順列/順列・組合せ/確率/関数の極限と微分法(松坂 和夫(著)、岩波書店)の第15章(「場合の数」 を数える - 順列・組合せ)、15.2(組合せ)、組合せの公式、問13、14、15、16、17、18.を取り組んでみる。

13. 
    $\left(\begin{array}{c}7\\ 3\end{array}\right)·\left(\begin{array}{c}5\\ 2\end{array}\right)=35·10=350$
14. 1. 
       $\left(\begin{array}{c}9\\ 4\end{array}\right)·\left(\begin{array}{c}5\\ 3\end{array}\right)·\left(\begin{array}{c}2\\ 2\end{array}\right)=1260$
    2. 
       $\left(\begin{array}{c}9\\ 3\end{array}\right)·\left(\begin{array}{c}6\\ 3\end{array}\right)·\left(\begin{array}{c}3\\ 3\end{array}\right)=84·20=1680$
    3. 
       $\frac{1680}{3!}=280$
15. 
    $\begin{array}{l}\left(\begin{array}{c}18\\ 6\end{array}\right)-\left(\begin{array}{c}12\\ 6\end{array}\right)·3=17·2·3·14·13-11·2·3·2·7·3\\ =6\left(17·14·13-11·2·7·3\right)\\ =6·14\left(17·13-11·3\right)\\ =84\left(221-33\right)\\ =84·188\\ =15792\end{array}$
16. 
    $\begin{array}{l}\left(\begin{array}{c}n\\ 2\end{array}\right)-n=\frac{n\left(n-1\right)-2n}{2}\\ =\frac{n\left(n-3\right)}{2}\end{array}$
17. 
    $\left(\begin{array}{c}5\\ 2\end{array}\right)·5+\left(\begin{array}{c}5\\ 1\end{array}\right)·\left(\begin{array}{c}5\\ 2\end{array}\right)+\left(\begin{array}{c}5\\ 3\end{array}\right)=11·10=110$
18. 
    $\left(\begin{array}{c}11\\ 7\end{array}\right)=\left(\begin{array}{c}11\\ 4\end{array}\right)=11·10·3=330$

コード(Emacs)

HTML5

<button id="run0">run</button>
<button id="clear0">clear</button>
<pre id="output0"></pre>
<script src="sample13.js"></script>

JavaScript

let btn0 = document.querySelector('#run0'),
    btn1 = document.querySelector('#clear0'),
    pre0 = document.querySelector('#output0');

let factorial = (n) => {
    return n <= 1 ? 1 : n * factorial(n - 1);
};

let combination = (n, r) => {
    return factorial(n) / (factorial(r) * factorial(n - r));
};

let output = () => {
    pre0.textContent =
        '13.\n' +
        `${combination(7, 3) * combination(5, 2)}\n` +
        '14-1.\n' +
        `${combination(9, 4) * combination(5, 3) * combination(2, 2)}\n` +
        '14-2.\n' +
        `${combination(9, 3) * combination(6, 3) * combination(3, 3)}\n` +
        '14-3.\n' +
        `${1680 / factorial(3)}\n` +
        '15.\n' +
        `${combination(18, 6) - combination(12, 6) * 3}\n` +
        '17.\n' +
        `${combination(5, 2) * 5 + combination(5, 1) * combination(5, 2) + combination(5, 3)}\n` +
        '18.\n' +
        `${combination(11, 7)}\n`;
};

btn0.onclick = output;
btn1.onclick = () => {
    pre0.textContent = '';
};

output();

13.
350
14-1.
1260
14-2.
1680
14-3.
280
15.
15792
17.
110
18.
330