{"id":9355,"date":"2025-08-15T17:38:19","date_gmt":"2025-08-15T12:08:19","guid":{"rendered":"https:\/\/namastedev.com\/blog\/?p=9355"},"modified":"2025-10-15T16:10:32","modified_gmt":"2025-10-15T10:40:32","slug":"combinations","status":"publish","type":"post","link":"https:\/\/namastedev.com\/blog\/combinations\/","title":{"rendered":"Combinations"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<!-- Evaluate Reverse Polish Notation 10-->\n<link\n    href=\"https:\/\/cdn.jsdelivr.net\/npm\/prismjs@1.29.0\/themes\/prism-tomorrow.min.css\"\n    rel=\"stylesheet\"\n\/>\n<script src=\"https:\/\/cdn.jsdelivr.net\/npm\/prismjs@1.29.0\/prism.min.js\"><\/script>\n<script src=\"https:\/\/cdn.jsdelivr.net\/npm\/prismjs@1.29.0\/plugins\/autoloader\/prism-autoloader.min.js\"><\/script>\n\n<style>\n.wp_blog_theme {\n  --primary: #E58C32;\n  --secondary: #030302;\n  --light-bg: #fef9f4;\n  --text-dark: #2d2d2d;\n  --tab-radius: 12px;\n  --shadow: 0 4px 12px rgba(0, 0, 0, 0.08);\n  --code-bg: #001f3f;\n  --code-text: #d4f1ff;\n}\n\n.wp_blog_container {\n  font-family: 'Segoe UI', sans-serif;\n  background: var(--light-bg);\n  margin: 0;\n  padding: 0;\n  color: var(--text-dark);\n}\n\n\/* Heading *\/\n.wp_blog_main-heading {\n  text-align: center;\n  font-size: 2.4rem;\n  color: var(--primary);\n  margin-top: 2.5rem;\n  font-weight: bold;\n}\n\n\/* Explanation Card *\/\n.wp_blog_explanation,\n.wp_blog_code-tabs-container {\n  max-width: 940px;\n  margin: 2rem auto;\n  padding: 2rem;\n  background: white;\n  border-radius: var(--tab-radius);\n  box-shadow: var(--shadow);\n}\n\n\/* Text and Visuals *\/\n.wp_blog_explanation h2 {\n  font-size: 1.4rem;\n  color: var(--primary);\n  margin-bottom: 0.5rem;\n}\n\n.wp_blog_explanation p,\n.wp_blog_explanation li {\n  font-size: 1.05rem;\n  line-height: 1.7;\n  margin: 0.5rem 0;\n}\n\n.wp_blog_explanation code {\n  background: #fef9f4; 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\/* makes it the reference for absolute children *\/\n}\n\n.wp_blog_toggle-btn {\n  position: absolute;\n  top: 1rem;\n  right: 1rem;\n  z-index: 9999;\n  padding: 0.5rem 0.8rem;\n  border-radius: 10%;\n  background: var(--primary);\n  color: white;\n  font-weight: bold;\n  cursor: pointer;\n  border: none;\n  box-shadow: var(--shadow);\n  transition: background 0.3s, transform 0.2s;\n}\n\n.wp_blog_toggle-btn:hover {\n  background: #cc772e;\n}\n\n.wp_blog_theme.dark-mode .wp_blog_code-tabs-container {\n  background: #1e1e1e;\n}\n<\/style>\n\n<div class=\"wp_blog_container wp_blog_theme\">\n      <button id=\"blogNotesThemeToggle\" class=\"wp_blog_toggle-btn\">\ud83c\udf19<\/button>\n<h1 class=\"wp_blog_main-heading\"><\/h1>\n\n<div class=\"wp_blog_explanation\">\n    <h2>Problem Statement:<\/h2>\n\n    <p>Given two integers <code>n<\/code> and <code>k<\/code>, return <i>all possible combinations of <code>k<\/code> numbers chosen from the range <code>[1, n]<\/code><\/i>.<\/p>\n    <p>\n        You may return the answer in <strong>any order<\/strong>.\n    <\/p>\n    <p><strong>Example 1:<\/strong><\/p>\n    <p><strong>Input:<\/strong> n = 4, k = 2<\/p>\n    <p><strong>Output:<\/strong><code> [[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]<\/code><\/p>\n    <p><strong>Explanation:<\/strong> There are 4 choose 2 = 6 total combinations.\nNote that combinations are unordered, i.e., [1,2] and [2,1] are considered to be the same combination.\n<\/p>\n\n    <p><strong>Example 2:<\/strong><\/p>\n    <p><strong>Input:<\/strong> n = 1, k = 1<\/p>\n    <p><strong>Output:<\/strong><code>[[1]]<\/code><\/p>\n    <p><strong>Explanation:<\/strong> There is 1 choose 1 = 1 total combination.<\/p>\n\n    <h2>Constraints:<\/h2>\n    <ul>\n        <li><code>1 <= n <= 20<\/code><\/li>\n        <li><code>1 <= k <= n<\/code><\/li>\n    <\/ul>\n\n    <h2>Approach:<\/h2>\n   <ul>\n        <li><strong>Generate all<\/strong> combinations of <code>size k<\/code> from numbers <strong>1 to n using recursion<\/strong>.<\/li>\n        <li><strong>Start from 1<\/strong> and explore each number by either including it or skipping it.<\/li>\n        <li><code>Keep adding numbers<\/code> to the <strong>current path<\/strong> until its length becomes k.<\/li>\n        <li>When <strong>path length<\/strong> is <code>k<\/code>, add it to the result.<\/li>\n        <li><strong>Backtrack<\/strong> by removing the last added number and try the next one.<\/li>\n    <\/ul> \n\n    <!-- <h2>Time Complexity:<\/h2>\n    <li><p><strong>Time Complexity = O(2<sup>n<\/sup>)<\/strong><\/p><\/li> \n    <h2>Space Complexity:<\/h2>\n    <li><p><strong>Space Complexity = O(2<sup>n<\/sup>)<\/strong><\/p><\/li> -->\n\n<h2>Dry Run<\/h2>\n<div style=\"background: var(--light-bg); border-left: 4px solid var(--primary); padding: 1rem; border-radius: var(--tab-radius); margin: 1rem 0; color: var(--text-dark);\">\n  \n  <p><strong>Input:<\/strong> <code>n = 4, k = 2<\/code><\/p>\n\n  <pre style=\"white-space: pre-wrap; background: var(--code-bg); padding: 1rem; border-radius: 8px; overflow-x: auto; color: var(--code-text);\">\nStep 0: Start Function combine(4, 2)\n\nInitialize:\nresult = []\npath = []\n\nCall backtrack([], 1)\n\nLoop i = 1\npath.push(1) \u2192 path = [1]\nCall backtrack([1], 2)\n\n  Loop i = 2\n  path.push(2) \u2192 path = [1, 2]\n  Call backtrack([1, 2], 3)\n  path.length == k \u2192 result = [[1, 2]]\n  path.pop() \u2192 path = [1]\n\n  Loop i = 3\n  path.push(3) \u2192 path = [1, 3]\n  Call backtrack([1, 3], 4)\n  path.length == k \u2192 result = [[1, 2], [1, 3]]\n  path.pop() \u2192 path = [1]\n\n  Loop i = 4\n  path.push(4) \u2192 path = [1, 4]\n  Call backtrack([1, 4], 5)\n  path.length == k \u2192 result = [[1, 2], [1, 3], [1, 4]]\n  path.pop() \u2192 path = [1]\n\npath.pop() \u2192 path = []\n\nLoop i = 2\npath.push(2) \u2192 path = [2]\nCall backtrack([2], 3)\n\n  Loop i = 3\n  path.push(3) \u2192 path = [2, 3]\n  Call backtrack([2, 3], 4)\n  path.length == k \u2192 result = [..., [2, 3]]\n  path.pop() \u2192 path = [2]\n\n  Loop i = 4\n  path.push(4) \u2192 path = [2, 4]\n  Call backtrack([2, 4], 5)\n  path.length == k \u2192 result = [..., [2, 4]]\n  path.pop() \u2192 path = [2]\n\npath.pop() \u2192 path = []\n\nLoop i = 3\npath.push(3) \u2192 path = [3]\nCall backtrack([3], 4)\n\n  Loop i = 4\n  path.push(4) \u2192 path = [3, 4]\n  Call backtrack([3, 4], 5)\n  path.length == k \u2192 result = [..., [3, 4]]\n  path.pop() \u2192 path = [3]\n\npath.pop() \u2192 path = []\n\nLoop i = 4\npath.push(4) \u2192 path = [4]\nCall backtrack([4], 5)\nLoop ends (start > n)\npath.pop() \u2192 path = []\n\nStep 3: End\nReturn result = [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]\n  <\/pre>\n\n  <p><strong>Output:<\/strong> \n    <code>[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]<\/code>\n  <\/p>\n<\/div>\n\n<\/div>\n<div class=\"wp_blog_code-tabs-container\">\n    <div class=\"wp_blog_code-tabs-header\">\n        <button class=\"wp_blog_code-tab-button active\" data-lang=\"js\">JavaScript<\/button>\n        <button class=\"wp_blog_code-tab-button\" data-lang=\"py\">Python<\/button>\n        <button class=\"wp_blog_code-tab-button\" data-lang=\"java\">Java<\/button>\n        <button class=\"wp_blog_code-tab-button\" data-lang=\"cpp\">C++<\/button>\n        <button class=\"wp_blog_code-tab-button\" data-lang=\"c\">C<\/button>\n        <button class=\"wp_blog_code-tab-button\" data-lang=\"cs\">C#<\/button>\n    <\/div>\n\n    <div class=\"wp_blog_code-tab-content active\" data-lang=\"js\">\n<pre><code class=\"language-javascript\">\nvar combine = function(n, k) {\n    let result = [];\n    let backtrack = (path, start) => {\n        if(path.length == k) {\n            result.push([...path]);\n            return;\n        }\n        for(let i = start; i <= n; i++){\n            path.push(i);\n            backtrack(path, i+1);\n            path.pop();\n        }\n    }\n    backtrack([], 1);\n    return result;\n};\n<\/code><\/pre>\n<\/div>\n\n<div class=\"wp_blog_code-tab-content\" data-lang=\"py\">\n<pre><code class=\"language-python\">\ndef backtrack(n, k, start, path, result):\n    if len(path) == k:\n        result.append(path[:])\n        return\n    for i in range(start, n + 1):\n        path.append(i)\n        backtrack(n, k, i + 1, path, result)\n        path.pop()\n\ndef combine(n, k):\n    result = []\n    backtrack(n, k, 1, [], result)\n    return result\n\n# Example\nprint(combine(4, 2))\n<\/code><\/pre>\n<\/div>\n\n<div class=\"wp_blog_code-tab-content\" data-lang=\"java\">\n<pre><code class=\"language-java\">\nimport java.util.*;\n\npublic class Main {\n    static void backtrack(int n, int k, int start, List<Integer> path, List<List<Integer>> result) {\n        if (path.size() == k) {\n            result.add(new ArrayList<>(path));\n            return;\n        }\n        for (int i = start; i <= n; i++) {\n            path.add(i);\n            backtrack(n, k, i + 1, path, result);\n            path.remove(path.size() - 1);\n        }\n    }\n\n    static List<List<Integer>> combine(int n, int k) {\n        List<List<Integer>> result = new ArrayList<>();\n        backtrack(n, k, 1, new ArrayList<>(), result);\n        return result;\n    }\n\n    public static void main(String[] args) {\n        List<List<Integer>> ans = combine(4, 2);\n        for (List<Integer> list : ans) {\n            System.out.println(list);\n        }\n    }\n}\n<\/code><\/pre>\n<\/div>\n\n<div class=\"wp_blog_code-tab-content\" data-lang=\"cpp\">\n<pre><code class=\"language-cpp\">\nvoid backtrack(int n, int k, int start, vector<int>& path, vector<vector<int>>& result) {\n    if (path.size() == k) {\n        result.push_back(path);\n        return;\n    }\n    for (int i = start; i <= n; i++) {\n        path.push_back(i);\n        backtrack(n, k, i + 1, path, result);\n        path.pop_back();\n    }\n}\nvector<vector<int>> combine(int n, int k) {\n    vector<vector<int>> result;\n    vector<int> path;\n    backtrack(n, k, 1, path, result);\n    return result;\n}\nint main() {\n    int n = 4, k = 2;\n    vector<vector<int>> ans = combine(n, k);\n    for (auto &vec : ans) {\n        for (int num : vec) cout << num << \" \";\n        cout << \"\\n\";\n    }\n    return 0;\n}\n<\/code><\/pre>\n<\/div>\n\n<div class=\"wp_blog_code-tab-content\" data-lang=\"c\">\n<pre><code class=\"language-c\">\nvoid backtrack(int n, int k, int start, int path[], int pathSize) {\n    if (pathSize == k) {\n        for (int i = 0; i < k; i++) printf(\"%d \", path[i]);\n        printf(\"\\n\");\n        return;\n    }\n    for (int i = start; i <= n; i++) {\n        path[pathSize] = i;\n        backtrack(n, k, i + 1, path, pathSize + 1);\n    }\n}\nint main() {\n    int n = 4, k = 2;\n    int path[20];\n    backtrack(n, k, 1, path, 0);\n    return 0;\n}\n<\/code><\/pre>\n<\/div>\n\n<div class=\"wp_blog_code-tab-content\" data-lang=\"cs\">\n<pre><code class=\"language-csharp\">\nusing System;\nusing System.Collections.Generic;\nclass Program {\n    static void Backtrack(int n, int k, int start, List<int> path, List<List<int>> result) {\n        if (path.Count == k) {\n            result.Add(new List<int>(path));\n            return;\n        }\n        for (int i = start; i <= n; i++) {\n            path.Add(i);\n            Backtrack(n, k, i + 1, path, result);\n            path.RemoveAt(path.Count - 1);\n        }\n    }\n    static List<List<int>> Combine(int n, int k) {\n        var result = new List<List<int>>();\n        Backtrack(n, k, 1, new List<int>(), result);\n        return result;\n    }\n    static void Main() {\n        var ans = Combine(4, 2);\n        foreach (var list in ans) {\n            Console.WriteLine(string.Join(\" \", list));\n        }\n    }\n}\n<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n<script>\ndocument.addEventListener(\"DOMContentLoaded\", () => {\n  const buttons = document.querySelectorAll(\".wp_blog_code-tab-button\");\n  const contents = document.querySelectorAll(\".wp_blog_code-tab-content\");\n\n  buttons.forEach((button) => {\n    button.addEventListener(\"click\", () => {\n      const lang = button.getAttribute(\"data-lang\");\n\n      buttons.forEach((btn) => btn.classList.remove(\"active\"));\n      contents.forEach((content) => content.classList.remove(\"active\"));\n\n      button.classList.add(\"active\");\n      document\n        .querySelector(`.wp_blog_code-tab-content[data-lang=\"${lang}\"]`)\n        .classList.add(\"active\");\n    });\n  });\n\n  const themeToggle = document.getElementById(\"blogNotesThemeToggle\");\n  const themeContainer = document.querySelector(\".wp_blog_theme\");\n\n  themeToggle.addEventListener(\"click\", () => {\n    themeContainer.classList.toggle(\"dark-mode\");\n    themeToggle.textContent =\n      themeContainer.classList.contains(\"dark-mode\") ? \"\u2600\ufe0f\" : \"\ud83c\udf19\";\n  });\n});\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>\ud83c\udf19 Problem Statement: Given two integers n and k, return all possible combinations of k numbers chosen from the range [1, n]. You may return the answer in any order. Example 1: Input: n = 4, k = 2 Output: [[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]] Explanation: There are 4 choose 2 = 6 total combinations. Note that combinations are<\/p>\n","protected":false},"author":108,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[210,322,260,176,175,211,811,810,174,172,173],"tags":[],"class_list":["post-9355","post","type-post","status-publish","format-standard","category-algorithms","category-algorithms-and-data-structures","category-c-c-plus-plus","category-csharp","category-cplusplus","category-data-structures","category-data-structures-and-algorithms","category-dsa","category-java","category-javascript","category-python"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/9355","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/users\/108"}],"replies":[{"embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/comments?post=9355"}],"version-history":[{"count":2,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/9355\/revisions"}],"predecessor-version":[{"id":10324,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/9355\/revisions\/10324"}],"wp:attachment":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/media?parent=9355"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/categories?post=9355"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/tags?post=9355"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}