数学 – JavaScript - 整数 - 集合(帰納法による等式の証明、階乗、組み合わせ)

代数系入門

学習環境/開発環境

• 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
• OS X El Capitan - Apple (OS)
• Emacs (Text Editor)
• JavaScript (プログラミング言語)
• JavaScript Library
  • kjs-array
• Safari(Web browser)
• 参考書籍
  • JavaScript 第6版 (David Flanagan(著)、村上 列(翻訳)、オライリージャパン)
  • JavaScriptリファレンス 第6版(David Flanagan(著)、木下 哲也(翻訳)、オライリージャパン)

代数系入門 (松坂 和夫(著)、岩波書店)の第1章(整数)、2(数学的帰納法と除法の定理)、問題4.を取り組んでみる。

問題4.

$\begin{array}{l}r=1\\ \left(\begin{array}{c}n\\ 0\end{array}\right)+\left(\begin{array}{c}n\\ 1\end{array}\right)=1+\frac{n!}{1!\left(n-1\right)!}=n\\ \left(\begin{array}{c}n+1\\ 1\end{array}\right)=\frac{\left(n+1\right)!}{1!\left(n+1-1\right)!}=n\\ \\ r=k\\ \left(\begin{array}{c}n\\ k-1\end{array}\right)+\left(\begin{array}{c}n\\ k\end{array}\right)=\left(\begin{array}{c}n+1\\ k\end{array}\right)\\ r=k+1\\ \left(\begin{array}{c}n\\ \left(k+1\right)-1\end{array}\right)+\left(\begin{array}{c}n\\ k+1\end{array}\right)\\ =\frac{n!}{k!\left(n-k\right)!}+\frac{n!}{\left(k+1\right)!\left(n-\left(k+1\right)\right)!}\\ =\frac{\left(k+1\right)n!+\left(n-k\right)n!}{\left(k+1\right)!\left(n-k\right)!}\\ =\frac{\left(n+1\right)!}{\left(k+1\right)!\left(n-k\right)!}\\ =\frac{n+1-k}{k+1}·\frac{\left(n+1\right)!}{k!\left(\left(n+1\right)-k\right)!}\\ =\frac{n+1-k}{k+1}·\left(\begin{array}{c}n+1\\ k\end{array}\right)\\ =\frac{n+1-k}{k+1}·\left(\left(\begin{array}{c}n\\ k-1\end{array}\right)+\left(\begin{array}{c}n\\ k\end{array}\right)\right)\\ =\frac{n+1-k}{k+1}·\left(\frac{n!}{\left(k-1\right)!\left(n-k+1\right)!}+\frac{n!}{k!\left(n-k\right)!}\right)\\ =\frac{n+1-k}{k+1}·\frac{kn!+\left(n-k+1\right)n!}{k!\left(n-k+1\right)!}\\ =\frac{\left(n+1\right)!}{\left(k+1\right)!\left(n-k\right)!}\\ =\frac{\left(n+1\right)!}{\left(k+1\right)!\left(\left(n+1\right)-\left(k+1\right)\right)!}\\ =\left(\begin{array}{c}n+1\\ k+1\end{array}\right)\end{array}$

JavaScript で確認。

HTML5

<label for="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="1">
<label for="r0">r = </label>
<input id="r0" type="number" min="1" step="1" value="1">
<div id="output0"></div>

<script src="array.js"></script>
<script src="sample4.js"></script>

JavaScript

コード(Emacs)

var input_n = document.querySelector('#n0'),
    input_r = document.querySelector('#r0'),
    inputs = [input_n, input_r],
    div_output = document.querySelector('#output0'),
    nl = '<br>',
    factorial,
    combination,
    output;

console.log(inputs);
factorial = function (n) {
    return Array.range(1, n + 1)
        .reduce(function (x, y) {
            return x * y;
        }, 1);
};
combination = function (n, r) {
    return factorial(n) / (factorial(r) * factorial(n - r));
};

output = function () {
    var left,
        right,
        n = parseInt(input_n.value, 10),
        r = parseInt(input_r.value, 10);

    if (1 <= r && r <= n) {
        left = combination(n, r - 1) + combination(n, r);
        right = combination(n + 1, r);
        div_output.innerHTML =
            '左辺 = ' + left + nl +
            '右辺 = ' + right + nl +
            '左辺 = 右辺 ' + (left === right) + nl +
            '|左辺 - 右辺| = ' + Math.abs(left - right) + nl;
    }
};

inputs.forEach(function (input) {
    input.onchange = output;
});

output();

n = r =