{"id":9444,"date":"2025-08-18T20:34:36","date_gmt":"2025-08-18T15:04:36","guid":{"rendered":"https:\/\/namastedev.com\/blog\/?p=9444"},"modified":"2025-10-15T16:32:43","modified_gmt":"2025-10-15T11:02:43","slug":"combination-sum-iii","status":"publish","type":"post","link":"https:\/\/namastedev.com\/blog\/combination-sum-iii\/","title":{"rendered":"Combination Sum III"},"content":{"rendered":"\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;   \/* light bg instead of dark blue *\/\n  color: #E58C32;        \/* brand orange *\/\n  padding: 3px 6px;\n  border-radius: 4px;\n  font-family: 'Courier New', monospace;\n  font-weight: 600;      \/* optional, makes it pop *\/\n}\n\n.wp_blog_explanation img {\n  max-width: 100%;\n  border-radius: var(--tab-radius);\n  margin-top: 1rem;\n  box-shadow: 0 2px 12px rgba(0, 0, 0, 0.06);\n}\n\n\/* Tab Buttons *\/\n.wp_blog_code-tabs-header {\n  display: flex;\n  flex-wrap: wrap;\n  gap: 0.5rem;\n  margin-bottom: 1rem;\n}\n\n.wp_blog_code-tab-button {\n  padding: 0.6rem 1.2rem;\n  border: 1px solid var(--primary);\n  background: white;\n  color: var(--primary);\n  border-radius: 50px;\n  font-weight: 600;\n  cursor: pointer;\n  transition: all 0.3s ease;\n}\n\n.wp_blog_code-tab-button:hover {\n  background: var(--secondary);\n}\n\n.wp_blog_code-tab-button.active {\n  background: var(--primary);\n  color: white;\n}\n\n\/* Code Content *\/\n.wp_blog_code-tab-content {\n  display: none;\n  background: var(--code-bg);\n  border-radius: var(--tab-radius);\n}\n\n.wp_blog_code-tab-content.active {\n  display: block;\n}\n\n.wp_blog_code-tab-content pre {\n  margin: 0;\n  padding: 1.5rem;\n  font-size: 1rem;\n  overflow-x: auto;\n  background: var(--code-bg);\n  border-radius: var(--tab-radius);\n  color: var(--code-text);\n}\n\n\/* Dark mode variables *\/\n.wp_blog_theme.dark-mode {\n  --light-bg: #121212;\n  --text-dark: #f5f5f5;\n  --shadow: 0 4px 12px rgba(255, 255, 255, 0.08);\n  --code-bg: #1e1e1e;\n  --code-text: #c5f0ff;\n}\n\n.wp_blog_theme.dark-mode .wp_blog_explanation {\n  background: #1e1e1e;\n}\n\n\/* Dark mode code highlight *\/\n.wp_blog_theme.dark-mode .wp_blog_explanation code {\n  background: #333;\n  color: #ffd27f;\n}\n\n.wp_blog_theme {\n  position: relative; \/* 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    <p>\n        Find all valid combinations of k numbers that sum up to n such that the following conditions are true:\n    <\/p>\n    \n    <ul>\n        <li>Only numbers <code>1<\/code> through <code>9<\/code> are used.<\/li>\n        <li>Each number is used at <strong>most once<\/strong>.<\/li>\n    <\/ul>\n\n    <p>Return a list of all possible valid combinations. The list must not contain the same combination twice, and the combinations may be returned in any order.<\/p>\n    \n    <p><strong>Example 1:<\/strong><\/p>\n    <p><strong>Input:<\/strong> k = 3, n = 7<\/p>\n    <p><strong>Output:<\/strong><code> [[1, 2, 4]]<\/code><\/p>\n    <p><strong>Explanation: <\/strong> <\/p>\n        <pre>\n        1 + 2 + 4 = 7 \n        <\/pre>\n       <p> There are no other valid combinations.<\/p>\n\n    <p><strong>Example 2:<\/strong><\/p>\n    <p><strong>Input:<\/strong> k = 3, n = 9<\/p>\n    <p><strong>Output:<\/strong><code> [[1,2,6],[1,3,5],[2,3,4]]<\/code><\/p>\n    <p><strong>Explanation: <\/strong> \n    <pre>\n    1 + 2 + 6 = 9\n    1 + 3 + 5 = 9\n    2 + 3 + 4 = 9\n    <\/pre>\n    There are no other valid combinations.<\/p>\n\n    <p><strong>Example 3:<\/strong><\/p>\n    <p><strong>Input:<\/strong> k = 4, n = 1<\/p>\n    <p><strong>Output:<\/strong><code> []<\/code><\/p>\n    <p><strong>Explanation: <\/strong>There are no valid combinations.\nUsing 4 different numbers in the range [1,9], the smallest sum we can get is 1+2+3+4 = 10 and since 10 > 1, there are no valid combination.<\/p>\n    <h2>Constraints:<\/h2>\n    <ul>\n        <li><code>2 <= k <= 9<\/code><\/li>\n        <li><code>1 <= n <= 60<\/code><\/li>\n    <\/ul>\n\n    <h2>Approach:<\/h2>\n    <ul>\n        <li>We need to find <strong>all combinations of k numbers<\/strong> (from 1 to 9) that <strong>sum up to n<\/strong>.<\/li>\n        <li><strong>Use backtracking<\/strong>:\n            <ul>\n                <li>Keep track of the current path (<code>path<\/code>) and remaining sum (<code>remainingSum<\/code>).<\/li>\n                <li>Stop if the path size reaches <code>k<\/code>. If <code>remainingSum == 0<\/code>, record the <strong>path<\/strong>.<\/li>\n                <li>Iterate numbers from start to 9 to avoid reuse and ensure ascending order.<\/li>\n                <li><strong>Add the number<\/strong>, recurse with updated sum and next start, then remove (backtrack).<\/li>\n            <\/ul>\n        <\/li>\n    <\/ul>\n\n    <!-- <h2>Time Complexity:<\/h2>\n    <li><p><strong>Time Complexity = O(2<sup>n<\/sup> * n)<\/strong><\/p><\/li> \n    <h2>Space Complexity:<\/h2>\n    <li><p><strong>Space Complexity =  O(2<sup>n<\/sup> * n)<\/strong> (output) + O(n) (stack)<\/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>k = 3, n = 7<\/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 combinationKSum(3, 7)\n\nInitialize:\nresult = []\npath = []\n\nCall backtrack(7, [], 1)\n\nLoop i = 1\npath.push(1) \u2192 path = [1]\nCall backtrack(6, [1], 2)\n\n  Loop i = 2\n  path.push(2) \u2192 path = [1, 2]\n  Call backtrack(4, [1, 2], 3)\n\n    Loop i = 3\n    path.push(3) \u2192 path = [1, 2, 3]\n    Call backtrack(1, [1, 2, 3], 4)\n      path.length == 3 but remainingSum = 1 \u2260 0 \u2192 return\n    path.pop() \u2192 path = [1, 2]\n\n    Loop i = 4\n    path.push(4) \u2192 path = [1, 2, 4]\n    Call backtrack(0, [1, 2, 4], 5)\n      path.length == 3 and remainingSum == 0 \u2192 result.push([1, 2, 4])\n    path.pop() \u2192 path = [1, 2]\n\n  Loop ends\n  path.pop() \u2192 path = [1]\n\n  Loop i = 3\n  path.push(3) \u2192 path = [1, 3]\n  Call backtrack(3, [1, 3], 4)\n\n    Loop i = 4\n    path.push(4) \u2192 path = [1, 3, 4]\n    Call backtrack(-1, [1, 3, 4], 5)\n      remainingSum < 0 \u2192 return\n    path.pop() \u2192 path = [1, 3]\n\n  Loop ends\n  path.pop() \u2192 path = [1]\n\n  Loop i = 4\n  path.push(4) \u2192 path = [1, 4]\n  Call backtrack(2, [1, 4], 5)\n\n    Loop i = 5\n    path.push(5) \u2192 path = [1, 4, 5]\n    Call backtrack(-3, [1, 4, 5], 6)\n      remainingSum < 0 \u2192 return\n    path.pop() \u2192 path = [1, 4]\n\n  Loop ends\n  path.pop() \u2192 path = [1]\n\nLoop i = 2\npath.push(2) \u2192 path = [2]\nCall backtrack(5, [2], 3)\n\n  Loop i = 3\n  path.push(3) \u2192 path = [2, 3]\n  Call backtrack(2, [2, 3], 4)\n\n    Loop i = 4\n    path.push(4) \u2192 path = [2, 3, 4]\n    Call backtrack(-2, [2, 3, 4], 5)\n      remainingSum < 0 \u2192 return\n    path.pop() \u2192 path = [2, 3]\n\n  Loop ends\n  path.pop() \u2192 path = [2]\n\n  Loop i = 4\n  path.push(4) \u2192 path = [2, 4]\n  Call backtrack(1, [2, 4], 5)\n\n    Loop i = 5\n    path.push(5) \u2192 path = [2, 4, 5]\n    Call backtrack(-4, [2, 4, 5], 6)\n      remainingSum < 0 \u2192 return\n    path.pop() \u2192 path = [2, 4]\n\n  Loop ends\n  path.pop() \u2192 path = [2]\n\nLoop ends\npath.pop() \u2192 path = []\n\nLoop i = 3, 4, 5 ... \u2192 no valid 3-number combos left within sum = 7\n\nStep 3: End\nReturn result = [[1, 2, 4]]\n  <\/pre>\n\n  <p><strong>Output:<\/strong> \n    <code>[[1, 2, 4]]<\/code>\n  <\/p>\n\n  <p><strong>Explanation:<\/strong> The only combination of <code>k = 3<\/code> numbers from <code>1..9<\/code> that sums to <code>7<\/code> is <code>[1, 2, 4]<\/code>. Each number is used at most once because recursion advances with <code>i+1<\/code>.<\/p>\n  \n<\/div>\n\n<\/div>\n\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 combinationSum3 = function(k, n) {\n    let result = [];\n    let backtrack = (remainingSum, path, start) => {\n        if(path.length == k){\n            if(remainingSum === 0){\n                result.push([...path]);\n            }\n            return;\n        }\n        for(let i=start; i<=9; i++){\n            path.push(i);\n            backtrack(remainingSum - i, path, i+1);\n            path.pop();\n        }\n    }\n    backtrack(n, [], 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 combinationSum3(k, n):\n    result = []\n    def backtrack(remainingSum, path, start):\n        if len(path) == k:\n            if remainingSum == 0:\n                result.append(path[:])\n            return\n        for i in range(start, 10):\n            path.append(i)\n            backtrack(remainingSum - i, path, i+1)\n            path.pop()\n    backtrack(n, [], 1)\n    return result\n\n# Example usage\nprint(combinationSum3(3, 7))\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.*;\nclass Solution {\n    public List<List<Integer>> combinationSum3(int k, int n) {\n        List<List<Integer>> result = new ArrayList<>();\n        backtrack(n, k, new ArrayList<>(), 1, result);\n        return result;\n    }\n    void backtrack(int remainingSum, int k, List<Integer> path, int start, List<List<Integer>> result) {\n        if (path.size() == k) {\n            if (remainingSum == 0) {\n                result.add(new ArrayList<>(path));\n            }\n            return;\n        }\n        for (int i = start; i <= 9; i++) {\n            path.add(i);\n            backtrack(remainingSum - i, k, path, i + 1, result);\n            path.remove(path.size() - 1);\n        }\n    }\n    public static void main(String[] args) {\n        Solution s = new Solution();\n        System.out.println(s.combinationSum3(3, 7));\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\">\nclass Solution {\npublic:\n    vector<vector<int>> combinationSum3(int k, int n) {\n        vector<vector<int>> result;\n        vector<int> path;\n        backtrack(n, k, path, 1, result);\n        return result;\n    }\n    void backtrack(int remainingSum, int k, vector<int>& path, int start, vector<vector<int>>& result) {\n        if(path.size() == k){\n            if(remainingSum == 0){\n                result.push_back(path);\n            }\n            return;\n        }\n        for(int i=start; i<=9; i++){\n            path.push_back(i);\n            backtrack(remainingSum - i, k, path, i+1, result);\n            path.pop_back();\n        }\n    }\n};\nint main(){\n    Solution s;\n    int k=3, n=7;\n    vector<vector<int>> ans = s.combinationSum3(k,n);\n    for(auto &vec : ans){\n        for(int x : vec) cout << x << \" \";\n        cout << \"\\n\";\n    }\n}\n<\/code><\/pre>\n<\/div>\n\n<div class=\"wp_blog_code-tab-content\" data-lang=\"c\">\n<pre><code class=\"language-c\">\nint result[100][10]; \nint resultSize = 0;\nint path[10];\nvoid backtrack(int remainingSum, int k, int pathSize, int start) {\n    if (pathSize == k) {\n        if (remainingSum == 0) {\n            for (int i = 0; i < pathSize; i++) {\n                result[resultSize][i] = path[i];\n            }\n            resultSize++;\n        }\n        return;\n    }\n    for (int i = start; i <= 9; i++) {\n        path[pathSize] = i;\n        backtrack(remainingSum - i, k, pathSize + 1, i + 1);\n    }\n}\nint main() {\n    int k = 3, n = 7;\n    backtrack(n, k, 0, 1);\n    for (int i = 0; i < resultSize; i++) {\n        for (int j = 0; j < k; j++) {\n            printf(\" %d \", result[i][j]);\n        }\n        printf(\"\\n\");\n    }\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\">\nclass Solution\n{\n    public IList<IList<int>> CombinationSum3(int k, int n)\n    {\n        var result = new List<IList<int>>();\n        Backtrack(n, k, new List<int>(), 1, result);\n        return result;\n    }\n    void Backtrack(int remainingSum, int k, List<int> path, int start, IList<IList<int>> result)\n    {\n        if (path.Count == k)\n        {\n            if (remainingSum == 0)\n            {\n                result.Add(new List<int>(path));\n            }\n            return;\n        }\n        for (int i = start; i <= 9; i++)\n        {\n            path.Add(i);\n            Backtrack(remainingSum - i, k, path, i + 1, result);\n            path.RemoveAt(path.Count - 1);\n        }\n    }\n    static void Main()\n    {\n        Solution s = new Solution();\n        var ans = s.CombinationSum3(3, 7);\n        foreach (var comb in ans)\n        {\n            Console.WriteLine(string.Join(\" \", comb));\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","protected":false},"excerpt":{"rendered":"<p>\ud83c\udf19 Problem Statement: Find all valid combinations of k numbers that sum up to n such that the following conditions are true: Only numbers 1 through 9 are used. Each number is used at most once. Return a list of all possible valid combinations. The list must not contain the same combination twice, and the<\/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":[210,322,176,175,211,811,810,174,172,173],"tags":[],"class_list":{"0":"post-9444","1":"post","2":"type-post","3":"status-publish","4":"format-standard","6":"category-algorithms","7":"category-algorithms-and-data-structures","8":"category-csharp","9":"category-cplusplus","10":"category-data-structures","11":"category-data-structures-and-algorithms","12":"category-dsa","13":"category-java","14":"category-javascript","15":"category-python"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/9444","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=9444"}],"version-history":[{"count":2,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/9444\/revisions"}],"predecessor-version":[{"id":10330,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/posts\/9444\/revisions\/10330"}],"wp:attachment":[{"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/media?parent=9444"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/categories?post=9444"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/namastedev.com\/blog\/wp-json\/wp\/v2\/tags?post=9444"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}