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href="/algorithm/pascal-s-triangle.html#pascal-s-triagle-basic-information" class="sidebar-link">Pascal's Triagle Basic Information</a></li><li class="sidebar-sub-header"><a href="/algorithm/pascal-s-triangle.html#implementation-of-pascal-s-triangle-in-c" class="sidebar-link">Implementation of Pascal's Triangle in C#</a></li></ul></li><li><a href="/algorithm/algo-print-a-m-n-matrix-in-square-wise.html" class="sidebar-link">Algo:- Print a m*n matrix in square wise</a></li><li><a href="/algorithm/matrix-exponentiation.html" class="sidebar-link">Matrix Exponentiation</a></li><li><a href="/algorithm/polynomial-time-bounded-algorithm-for-minimum-vertex-cover.html" class="sidebar-link">polynomial-time bounded algorithm for Minimum Vertex Cover</a></li><li><a href="/algorithm/dynamic-time-warping.html" class="sidebar-link">Dynamic Time Warping</a></li><li><a href="/algorithm/fast-fourier-transform.html" class="sidebar-link">Fast Fourier Transform</a></li><li><a href="/algorithm/pseudocode.html" class="sidebar-link">Pseudocode</a></li><li><a href="/algorithm/contributors.html" class="sidebar-link">The Contributors</a></li></ul></section></li></ul> </aside> <main class="page"> <div class="theme-default-content content__default"><h1 id="pascal-s-triangle"><a href="#pascal-s-triangle" class="header-anchor">#</a> Pascal's Triangle</h1> <h2 id="pascal-triangle-in-c"><a href="#pascal-triangle-in-c" class="header-anchor">#</a> Pascal triangle in C</h2> <div class="language-cpp extra-class"><pre class="language-cpp"><code><span class="token keyword">int</span> i<span class="token punctuation">,</span> space<span class="token punctuation">,</span> rows<span class="token punctuation">,</span> k<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">,</span> count <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">,</span> count1 <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
row<span class="token operator">=</span><span class="token number">5</span><span class="token punctuation">;</span>
<span class="token keyword">for</span><span class="token punctuation">(</span>i<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span> i<span class="token operator"><=</span>rows<span class="token punctuation">;</span> <span class="token operator">++</span>i<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">for</span><span class="token punctuation">(</span>space<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span> space <span class="token operator"><=</span> rows<span class="token operator">-</span>i<span class="token punctuation">;</span> <span class="token operator">++</span>space<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">" "</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token operator">++</span>count<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">while</span><span class="token punctuation">(</span>k <span class="token operator">!=</span> <span class="token number">2</span><span class="token operator">*</span>i<span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">if</span> <span class="token punctuation">(</span>count <span class="token operator"><=</span> rows<span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">"%d "</span><span class="token punctuation">,</span> i<span class="token operator">+</span>k<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token operator">++</span>count<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">else</span>
<span class="token punctuation">{</span>
<span class="token operator">++</span>count1<span class="token punctuation">;</span>
<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">"%d "</span><span class="token punctuation">,</span> <span class="token punctuation">(</span>i<span class="token operator">+</span>k<span class="token operator">-</span><span class="token number">2</span><span class="token operator">*</span>count1<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token operator">++</span>k<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
count1 <span class="token operator">=</span> count <span class="token operator">=</span> k <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">"\n"</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
</code></pre></div><p><strong>Output</strong></p> <div class="language- extra-class"><pre class="language-text"><code>
1
2 3 2
3 4 5 4 3
4 5 6 7 6 5 4
5 6 7 8 9 8 7 6 5
</code></pre></div><h2 id="pascal-s-triagle-basic-information"><a href="#pascal-s-triagle-basic-information" class="header-anchor">#</a> Pascal's Triagle Basic Information</h2> <p>One of the most interesting Number Patterns is <a href="https://en.wikipedia.org/wiki/Pascal%27s_triangle" target="_blank" rel="noopener noreferrer">Pascal's Triangle<span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a>. The Name "Pascal's Triangle" named after <a href="https://en.wikipedia.org/wiki/Blaise_Pascal" target="_blank" rel="noopener noreferrer">Blaise Pascal<span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a>, a famous French Mathematician and Philosopher.</p> <p>In Mathematics, Pascal's Triangle is a triangular array of binomial coefficients.The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows.</p> <p><strong>The triangle is constructed in the below manner:</strong></p> <ul><li>In the topmost row, there is a unique nonzero entry 1.</li> <li>Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0.</li></ul> <p>For example, the initial number in the first (or any other) row is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.</p> <p><strong>Equation to generate each entry in Pascal triangle:</strong></p> <p><a href="https://i.stack.imgur.com/7WEJ9.png" target="_blank" rel="noopener noreferrer"><img src="https://i.stack.imgur.com/7WEJ9.png" alt="Pascal Equation"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p>for any non-negative integer n and any integer k between 0 and n, inclusive. This recurrence for the binomial coefficients is known as <a href="https://en.wikipedia.org/wiki/Pascal%27s_rule" target="_blank" rel="noopener noreferrer">Pascal's rule<span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a>. Pascal's triangle has higher dimensional generalizations. The three-dimensional version is called Pascal's pyramid or Pascal's tetrahedron, while the general versions are called Pascal's simplices.</p> <p><strong>Example of Pascal's Triangle:</strong></p> <p><a href="https://i.stack.imgur.com/chhsE.gif" target="_blank" rel="noopener noreferrer"><img src="https://i.stack.imgur.com/chhsE.gif" alt="Pascal's Triangle Example"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <h2 id="implementation-of-pascal-s-triangle-in-c"><a href="#implementation-of-pascal-s-triangle-in-c" class="header-anchor">#</a> Implementation of Pascal's Triangle in C#</h2> <div class="language-cpp extra-class"><pre class="language-cpp"><code><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">PascalsTriangle</span>
<span class="token punctuation">{</span>
<span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">PascalTriangle</span><span class="token punctuation">(</span><span class="token keyword">int</span> n<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> line <span class="token operator">=</span> <span class="token number">1</span><span class="token punctuation">;</span> line <span class="token operator"><=</span> n<span class="token punctuation">;</span> line<span class="token operator">++</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">int</span> c <span class="token operator">=</span> <span class="token number">1</span><span class="token punctuation">;</span>
<span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">1</span><span class="token punctuation">;</span> i <span class="token operator"><=</span> line<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
Console<span class="token punctuation">.</span><span class="token function">WriteLine</span><span class="token punctuation">(</span>c<span class="token punctuation">)</span><span class="token punctuation">;</span>
c <span class="token operator">=</span> c <span class="token operator">*</span> <span class="token punctuation">(</span>line <span class="token operator">-</span> i<span class="token punctuation">)</span> <span class="token operator">/</span> i<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
Console<span class="token punctuation">.</span><span class="token function">WriteLine</span><span class="token punctuation">(</span><span class="token string">"\n"</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">int</span> <span class="token function">Main</span><span class="token punctuation">(</span><span class="token keyword">int</span> input<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token function">PascalTriangle</span><span class="token punctuation">(</span>input<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">return</span> input<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</code></pre></div></div> <footer class="page-edit"><div class="edit-link"><a href="https://github.com/devtut/generate/edit/master/docs/algorithm/pascal-s-triangle.md" target="_blank" rel="noopener noreferrer">Edit this page on GitHub</a> <span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></div> <!----></footer> <div class="page-nav"><p class="inner"><span class="prev">
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