<p>Given a string <code>num</code> that contains only digits and an integer <code>target</code>, return <em><strong>all possibilities</strong> to insert the binary operators </em><code>&#39;+&#39;</code><em>, </em><code>&#39;-&#39;</code><em>, and/or </em><code>&#39;*&#39;</code><em> between the digits of </em><code>num</code><em> so that the resultant expression evaluates to the </em><code>target</code><em> value</em>.</p>

<p>Note that operands in the returned expressions <strong>should not</strong> contain leading zeros.</p>

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

<pre>
<strong>Input:</strong> num = &quot;123&quot;, target = 6
<strong>Output:</strong> [&quot;1*2*3&quot;,&quot;1+2+3&quot;]
<strong>Explanation:</strong> Both &quot;1*2*3&quot; and &quot;1+2+3&quot; evaluate to 6.
</pre>

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

<pre>
<strong>Input:</strong> num = &quot;232&quot;, target = 8
<strong>Output:</strong> [&quot;2*3+2&quot;,&quot;2+3*2&quot;]
<strong>Explanation:</strong> Both &quot;2*3+2&quot; and &quot;2+3*2&quot; evaluate to 8.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> num = &quot;3456237490&quot;, target = 9191
<strong>Output:</strong> []
<strong>Explanation:</strong> There are no expressions that can be created from &quot;3456237490&quot; to evaluate to 9191.
</pre>

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

<ul>
	<li><code>1 &lt;= num.length &lt;= 10</code></li>
	<li><code>num</code> consists of only digits.</li>
	<li><code>-2<sup>31</sup> &lt;= target &lt;= 2<sup>31</sup> - 1</code></li>
</ul>

class Solution:
    def addOperators(self, num: str, target: int) -> List[str]:
        self.cache = dict()
        arrangements = self.makeRec(num, False)
        result = []
        for arrangement in arrangements:
            if eval(arrangement) == target:
                result.append(arrangement)
        return result
    def makeRec(self, num: str, allowLeadingZeros: bool) -> List[str]:
        if len(num) == 1:
            return [num]
        if (num, allowLeadingZeros) not in self.cache:
            result = []
            for recursiveArrangement in self.makeRec(num[1:], False):
                result.append(num[0] + "+" + recursiveArrangement)
                result.append(num[0] + "-" + recursiveArrangement)
                result.append(num[0] + "*" + recursiveArrangement)
            if num[0] != "0" or allowLeadingZeros:
                for recursiveArrangement in self.makeRec(num[1:], True):
                    result.append(num[0] + recursiveArrangement)
            self.cache[(num, allowLeadingZeros)] = result
        return self.cache[(num, allowLeadingZeros)]
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