<p>We call a positive integer <strong>special</strong> if all of its digits are <strong>distinct</strong>.</p>

<p>Given a <strong>positive</strong> integer <code>n</code>, return <em>the number of special integers that belong to the interval </em><code>[1, n]</code>.</p>

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

<pre>
<strong>Input:</strong> n = 20
<strong>Output:</strong> 19
<strong>Explanation:</strong> All the integers from 1 to 20, except 11, are special. Thus, there are 19 special integers.
</pre>

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

<pre>
<strong>Input:</strong> n = 5
<strong>Output:</strong> 5
<strong>Explanation:</strong> All the integers from 1 to 5 are special.
</pre>

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

<pre>
<strong>Input:</strong> n = 135
<strong>Output:</strong> 110
<strong>Explanation:</strong> There are 110 integers from 1 to 135 that are special.
Some of the integers that are not special are: 22, 114, and 131.</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 2 * 10<sup>9</sup></code></li>
</ul>

class Solution:
    def countSpecialNumbers(self, n: int) -> int:
        cache={}
        s=str(n)
        def dfs(index,mask,hasLeadingZeroes,hasUpperLimit):
            if index==len(s):
                if hasLeadingZeroes:
                    return 0
                return 1

            res=0
            if (index,mask,hasLeadingZeroes,hasUpperLimit) in cache:
                return cache[(index,mask,hasLeadingZeroes,hasUpperLimit)]

            if hasLeadingZeroes:
                res+=dfs(index+1,mask,hasLeadingZeroes,False)
            
            low= 1 if hasLeadingZeroes else 0
            high= int(s[index]) if hasUpperLimit else 9

            for i in range(low,high+1):
                if not mask & 1<<i:
                    res+=dfs(index+1,mask|1<<i,False,True if hasUpperLimit and i==high else False)
            
            cache[(index,mask,hasLeadingZeroes,hasUpperLimit)]=res

            return res

        return dfs(0,0,True,True)

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