<p>You are given an integer array <code>cards</code> of length <code>4</code>. You have four cards, each containing a number in the range <code>[1, 9]</code>. You should arrange the numbers on these cards in a mathematical expression using the operators <code>[&#39;+&#39;, &#39;-&#39;, &#39;*&#39;, &#39;/&#39;]</code> and the parentheses <code>&#39;(&#39;</code> and <code>&#39;)&#39;</code> to get the value 24.</p>

<p>You are restricted with the following rules:</p>

<ul>
	<li>The division operator <code>&#39;/&#39;</code> represents real division, not integer division.

	<ul>
		<li>For example, <code>4 / (1 - 2 / 3) = 4 / (1 / 3) = 12</code>.</li>
	</ul>
	</li>
	<li>Every operation done is between two numbers. In particular, we cannot use <code>&#39;-&#39;</code> as a unary operator.
	<ul>
		<li>For example, if <code>cards = [1, 1, 1, 1]</code>, the expression <code>&quot;-1 - 1 - 1 - 1&quot;</code> is <strong>not allowed</strong>.</li>
	</ul>
	</li>
	<li>You cannot concatenate numbers together
	<ul>
		<li>For example, if <code>cards = [1, 2, 1, 2]</code>, the expression <code>&quot;12 + 12&quot;</code> is not valid.</li>
	</ul>
	</li>
</ul>

<p>Return <code>true</code> if you can get such expression that evaluates to <code>24</code>, and <code>false</code> otherwise.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> cards = [4,1,8,7]
<strong>Output:</strong> true
<strong>Explanation:</strong> (8-4) * (7-1) = 24
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> cards = [1,2,1,2]
<strong>Output:</strong> false
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>cards.length == 4</code></li>
	<li><code>1 &lt;= cards[i] &lt;= 9</code></li>
</ul>

class Solution {
public:
    double eps=1e-9;
    bool ans=false;
    vector<double> validnums(double a, double b){
        vector<double> valid;
        valid.push_back(a+b);
        valid.push_back(a-b);
        valid.push_back(b-a);
        valid.push_back(a*b);
        if(b>eps) valid.push_back(a/b);
        if(a>eps) valid.push_back(b/a);
        return valid;
    }

    void f(vector<double> &nums){
        int n=nums.size();
        if(n==1){
            if(abs(24.0-nums[0])<eps) ans=true;
            return;
        }
        for(int i=0; i<n; i++){
            for(int j=0; j<n; j++){
                if(i==j) continue;
                vector<double> upd;
                for(int k=0; k<n; k++) if(k!=i && k!=j) upd.push_back(nums[k]);
                vector<double> psblnums;
                psblnums=validnums(nums[i],nums[j]);
                for(auto num:psblnums){
                    upd.push_back(num);
                    f(upd);
                    if(ans) return;
                    upd.pop_back();
                } 
            }
        }
    }

    bool judgePoint24(vector<int>& cards) {
        vector<double> nums(cards.begin(),cards.end());
        f(nums);
        return ans;
    }
};
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