数学 - JavaScript - 「場合の数」 を数える - 順列・組合せ – 二項定理 - 二項定理の応用、二項係数の性質(等式の証明)

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)、二項定理の応用、二項係数の性質、問40、41、42.を取り組んでみる。

40. 1. 
       $\begin{array}{l}r·\frac{n!}{\left(n-r\right)!r!}=\frac{n!}{\left(n-r\right)!\left(r-1\right)!}\\ n·\frac{\left(n-1\right)!}{\left(n-1-r+1\right)!\left(r-1\right)!}=\frac{n!}{\left(n-r\right)!\left(r-1\right)!}\end{array}$
    2. 
       $\begin{array}{l}{\displaystyle \sum _{i=1}^{n}i\left(\begin{array}{c}n\\ i\end{array}\right)}\\ ={\displaystyle \sum _{i=1}^{n}i\frac{n!}{\left(n-i\right)!i!}}\\ =n{\displaystyle \sum _{i=1}^n\frac{\left(n-1\right)!}{\left(n-i\right)!\left(i-1\right)!}}\\ =n{\displaystyle \sum _{i=0}^{n-1}\frac{\left(n-1\right)!}{\left(n-\left(i-1\right)\right)!i!}}\\ =n{\displaystyle \sum _{i=0}^{n-1}\frac{\left(n-1\right)!}{\left(\left(n-1\right)-i\right)!i!}}\\ =n{\displaystyle \sum _{i=0}^{n-1}\left(\begin{array}{c}n\\ i\end{array}\right)}\\ =n{\left(1+1\right)}^{n-1}\\ =n·2^{n-1}\end{array}$
41. 
    $\left(\begin{array}{c}m+n\\ r\end{array}\right)x^r={\displaystyle \sum _{i=0}^r\left(\begin{array}{c}m\\ i\end{array}\right)\left(\begin{array}{c}n\\ r-i\end{array}\right)}$
42. 
    $\begin{array}{l}\frac{\left(\begin{array}{c}n\\ r+1\end{array}\right)}{\left(\begin{array}{c}n\\ r\end{array}\right)}=\frac{n!}{\left(n-r-1\right)!\left(r+1\right)!}·\frac{\left(n-r\right)!r!}{n!}\\ =\frac{n-r}{r+1}\\ \\ \frac{n-r}{r+1}=1\\ n-r=r+1\\ r=\frac{n-1}{2}\\ \\ r<\frac{n-1}{2}\\ \left(\begin{array}{c}n\\ r\end{array}\right)<\left(\begin{array}{c}n\\ r+1\end{array}\right)\\ r>\frac{n-1}{2}\\ \left(\begin{array}{c}n\\ r\end{array}\right)>\left(\begin{array}{c}n\\ r+1\end{array}\right)\end{array}$

コード(Emacs)

HTML5

<button id="run0">run</button>
<button id="clear0">clear</button>
<pre id="output0"></pre>
<script src="sample40.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 term = (n, i) => i * combination(n, i),
    f = (n) => range(1, n + 1).map((i) => term(n, i)).reduce((x, y) => x + y);


let output = () => {        
    p('40-1.');
    let r = Math.floor(Math.random() * 100) + 1,
        n = r + Math.floor(Math.random()) + 1;
    
    p(`${r * combination(n, r)} === ${n * combination(n - 1, r - 1)}: ` +
      `${r * combination(n, r) === n * combination(n - 1, r - 1)}`);

    p('40-2.');
    n = Math.floor(Math.random() * 100);
    p(`${n * Math.pow(2, n - 1)} === ${f(n)}: ` +
      `${n * Math.pow(2, n - 1) === f(n)}`);

    p('42.');
    n = Math.floor(Math.random() * 100) + 2;
    p(range(0, n + 1).map((i) => Math.floor(combination(n, i))));
};

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

output();

40-1.
462 === 462: true
40-2.
7784628224 === 7784628224: true
42.
1,58,1653,30856,424270,4582116,40475358,300674088,1916797311,10648873950,52179482355,227692286640,891794789340,3155581562280,10142940735900,29752626158640,79960182801345,197548686920970,449972009097765,947309492837400,1847253511032930,3342649210440541,5621728217559091,8799226775309881,12832205713993578,17451799771031264,22150361247847372,26252279997448744,29065024282889680,30067266499541052,29065024282889680,26252279997448744,22150361247847372,17451799771031264,12832205713993578,8799226775309881,5621728217559091,3342649210440541,1847253511032930,947309492837400,449972009097765,197548686920970,79960182801345,29752626158640,10142940735900,3155581562280,891794789340,227692286640,52179482355,10648873950,1916797311,300674088,40475358,4582116,424270,30856,1653,58,1