数学 - JavaScript - 「場合の数」 を数える - 順列・組合せ – 二項定理 - 多項定理(項の種類、同じ型、係数)

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)、15.3(二項定理)、多項定理、問46.を取り組んでみる。

46. 
    $\begin{array}{l}\left(\begin{array}{c}4+6-1\\ 6\end{array}\right)=\left(\begin{array}{c}9\\ 6\end{array}\right)=\left(\begin{array}{c}9\\ 3\end{array}\right)=3·4·7=84\\ \\ 6+0+0+0=6,\left(\begin{array}{c}6\\ 0\end{array}\right)=1,a^9\\ 5+1+0+0=6,\left(\begin{array}{c}6\\ 1\end{array}\right)=6,a^{9}b\\ 4+2+0+0=6,\left(\begin{array}{c}6\\ 2\end{array}\right)=15,a^4{b}^2\\ 4+1+1+0=6,\left(\begin{array}{c}6\\ 4\end{array}\right)·\left(\begin{array}{c}2\\ 1\end{array}\right)=30,a^{4}bc\\ 3+3+0+0=6,\left(\begin{array}{c}6\\ 3\end{array}\right)=20,a^3{b}^3\\ 3+2+1+0=6,\left(\begin{array}{c}6\\ 3\end{array}\right)·\left(\begin{array}{c}3\\ 2\end{array}\right)=60,a^3{b}^{2}c\\ 3+1+1+1=6,\left(\begin{array}{c}6\\ 3\end{array}\right)·\left(\begin{array}{c}3\\ 1\end{array}\right)·\left(\begin{array}{c}2\\ 1\end{array}\right)=120,a^{3}bcd\\ 2+2+2+0=6,\left(\begin{array}{c}6\\ 2\end{array}\right)·\left(\begin{array}{c}4\\ 2\end{array}\right)=15·6=90,a^2{b}^2{c}^2\\ 2+2+1+1=6,\left(\begin{array}{c}6\\ 2\end{array}\right)·\left(\begin{array}{c}4\\ 2\end{array}\right)·\left(\begin{array}{c}2\\ 1\end{array}\right)=180,a^2{b}^{2}cd\end{array}$

コード(Emacs)

HTML5

<button id="run0">run</button>
<button id="clear0">clear</button>
<pre id="output0"></pre>
<script src="sample46.js"></script>

JavaScript

let btn0 = document.querySelector('#run0'),
    btn1 = document.querySelector('#clear0'),
    pre0 = document.querySelector('#output0'),
    p = (x) => pre0.textContent += x + '\n';

let range = (start, end, step=1) => {
    let iter = (i, result) => {
        return i >= end ? result : iter(i + step, result.concat([i]));
    }
    return iter(start, []);
};
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 = () => {        
    p('46.');
    [
        [[9, 3]],
        [[6, 0]],
        [[6, 1]],
        [[6, 2]],
        [[6, 4], [2, 1]],
        [[6, 3]],
        [[6, 3], [3, 2]],
        [[6, 3], [3, 1], [2, 1]],
        [[6, 2], [4, 2]],
        [[6, 2], [4, 2], [2, 1]]
    ].forEach((args) => {
        p(args.reduce((prev, x) => prev * combination(...x), 1));
    });
};

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

output();