数学 - JavaScript - 「場合の数」 を数える - 順列・組合せ – 二項定理 - 多項定理(多項係数)

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43、44、45.を取り組んでみる。

43. 1. 
       $\frac{7!}{2!1!4!}=105$
    2. 
       $\frac{7!}{0!3!4!}=35$
    3. 
       $\frac{6!}{1!2!3!0!}=60$
    4. 
       $\frac{6!}{2!2!1!1!}=180$
44. 
    $\begin{array}{l}\frac{6!}{p!q!r!}{\left(x\right)}^p{\left(-2y\right)}^q{5}^r\\ =\frac{6!}{p!q!r!}{\left(-2\right)}^q{5}^r·x^p{y}^q\\ \\ p=2,q=4,r=6-2-4=0\\ \frac{6!}{2!4!0!}·{\left(-2\right)}^4{5}^0=15·16=240\\ \\ p=2,q=1,r=6-p-q=3\\ \frac{6!}{2!1!3!}{\left(-2\right)}^1{5}^3=-60·250=-15000\\ \\ p=1,q=1,r=6-1-1=4\\ \frac{6!}{1!1!4!}{\left(-2\right)}^1{5}^4=-30·2·625=-37500\end{array}$
45. 
    $\begin{array}{l}\frac{7!}{p!q!r!}{\left(x^2\right)}^p{\left(-3x\right)}^q{4}^r\\ =\frac{7!}{p!q!r!}{\left(-3\right)}^q{4}^r{x}^{2p+q}\\ \\ 2p+q=2\\ p=0,q=2,r=5\\ p=1,q=0,r=6\\ \\ \frac{7!}{0!2!5!}{\left(-3\right)}^2{4}^5+\frac{7!}{1!0!6!}{\left(-3\right)}^0{4}^6\\ =21·9·4^5+7·4^6\\ =4^5\left(189+28\right)\\ =1024·217\\ =222208\\ \\ 2p+q=5\\ p=0,q=5,r=2\\ p=1,q=3,r=3\\ p=2,q=1,r=4\\ \frac{7!}{0!5!2!}{\left(-3\right)}^5{4}^2+\frac{7!}{1!3!3!}{\left(-3\right)}^3{4}^3+\frac{7!}{2!1!4!}{\left(-3\right)}^1{4}^4\\ =21{\left(-3\right)}^5{4}^2+140{\left(-3\right)}^3{4}^3+105\left(-3\right)4^4\\ =\left(-3\right)4^2\left(21·{\left(-3\right)}^4+140{\left(-3\right)}^2·4+105·4^2\right)\\ =\text{}\text{}-48\left(1701+5040+1680\right)\\ =-404208\\ \\ 2p+q=10\\ p=3,q=4,r=0\\ p=4,q=2,r=1\\ p=5,q=0,r=2\\ \frac{7!}{3!4!0!}{\left(-3\right)}^4{4}^0+\frac{7!}{4!2!1!}{\left(-3\right)}^2{4}^1+\frac{7!}{5!0!2!}{\left(-3\right)}^0{4}^2\\ =35·81+105·9·4+21·16\\ =2835+3780+336\\ =6951\end{array}$

コード(Emacs)

HTML5

<button id="run0">run</button>
<button id="clear0">clear</button>
<pre id="output0"></pre>
<script src="sample43.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 multinomialCoefficient = (n, ...args) => {
    return factorial(n) / args.map(factorial).reduce((x, y) => x * y);
};

let output = () => {        
    p('43-1.');
    p(multinomialCoefficient(7, 2, 1, 4));
    p('43-2.');
    p(multinomialCoefficient(7, 1, 3, 4));
    p('43-3.');
    p(multinomialCoefficient(6, 1, 2, 3));
    p('43-4.');
    p(multinomialCoefficient(6, 2, 2, 1, 1));
    p('44.');
    p(multinomialCoefficient(6, 2, 4, 0) * Math.pow(-2, 4))
    p(multinomialCoefficient(6, 2, 1, 3) * -2 * Math.pow(5, 3));
    p(multinomialCoefficient(6, 1, 1, 4) * -2 * Math.pow(5, 4));
    p('45.');
    p(multinomialCoefficient(7, 0, 2, 5) * Math.pow(-3, 2) * Math.pow(4, 5) +
      multinomialCoefficient(7, 1, 0, 6) * 1 * Math.pow(4, 6));
    p(multinomialCoefficient(7, 0, 5, 2) * Math.pow(-3, 5) * Math.pow(4, 2) +
      multinomialCoefficient(7, 1, 3, 3) * Math.pow(-3, 3) * Math.pow(4, 3) +
      multinomialCoefficient(7, 2, 1, 4) * Math.pow(-3, 1) * Math.pow(4, 4));
    p(multinomialCoefficient(7, 3, 4, 0) * Math.pow(-3, 4)  +
      multinomialCoefficient(7, 4, 2, 1) * Math.pow(-3, 2) * 4 +
      multinomialCoefficient(7, 5, 0, 2) * Math.pow(4, 2));
    
};

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

output();

43-1.
105
43-2.
35
43-3.
60
43-4.
180
44.
240
-15000
-37500
45.
222208
-404208
6951