<PRE><B>Question 1135742</B>
&lt;br&gt;
You have received an excellent response from tutor @Math_Helper, showing how to find the answer to the question by finding and following a pattern.&lt;br&gt;
I followed a similar but different path to the answer, using a different kind of pattern.&lt;br&gt;
Both solutions are good examples of how you can solve many problems by looking for patterns.&lt;br&gt;
My path to the answer started with listing all the 4-digit numbers with leading digit 3 that meet the requirements, then listing the ones with 4 as the leading digit, and so on.  Here is what I found.&lt;br&gt;&lt;pre&gt;
1: leading digit 3&lt;br&gt;
There is clearly only 1 4-digit number with leading digit 3 that meets the requirements: 3210&lt;br&gt;
2: leading digit 4&lt;br&gt;
The 4-digit numbers with leading digit 4 that meet the requirements can be separated into two distinct groups:
(a) All the 4-digit numbers with leading digit 3 that meet the requirements, with the leading digit &quot;3&quot; replaced by leading digit &quot;4&quot;; and
(b) All the 4-digit numbers that meet the requirements having first TWO digits &quot;43&quot;.&lt;br&gt;
From (a) we get the single 4-digit number 4210.&lt;br&gt;
From (b) we get...
4321, 4320;  (2 numbers)
4310  (1 number)&lt;br&gt;
We have 1 4-digit number from (a) and (2+1) = 3 from (b); the total number of 4-digit numbers that meet the requirements is 1+3 = 4.&lt;br&gt;
3: leading digit 5&lt;br&gt;
The 4-digit numbers with leading digit 5 that meet the requirements can be separated into two distinct groups:
(a) All the 4-digit numbers with leading digit 4 that meet the requirements, with the leading digit &quot;4&quot; replaced by leading digit &quot;5&quot;; and
(b) All the 4-digit numbers that meet the requirements having first TWO digits &quot;54&quot;.&lt;br&gt;
From (a) we get 4 numbers that meet the requirements.&lt;br&gt;
From (b) we get...
5432, 5431, 5430;  (3 numbers)
5421, 5420;  (2 numbers)
5410  (1 number)
&lt;br&gt;
We have 4 4-digit number from (a) and (3+2+1) = 6 from (b); the total number of 4-digit numbers that meet the requirements is 4+6 = 10.&lt;br&gt;
4: leading digit 6 or higher&lt;br&gt;
I won&#39;t go into the details for any of the other cases; the pattern is clear to me.  If you don&#39;t see it yet, go through the details for one or two more leading digits to help you.&lt;br&gt;
The results you find are...&lt;br&gt;
leading digit 6: from(a), the 10 numbers from the case with leading digit 5; and from (b), 4+3+2+1 = 10 numbers.  Total: 10+10 = 20 numbers.&lt;br&gt;
leading digit 7: from (a), the 20 numbers from the case with leading digit 6; and from (b), 5+4+3+2+1 = 15 numbers.  Total 20+15 = 35 numbers.&lt;br&gt;
leading digit 8: from (a), the 35 numbers from the case with leading digit 7; and from (b), 6+5+4+3+2+1 = 21 numbers.  Total 35+21 = 56 numbers.&lt;br&gt;
leading digit 9: from (a), the 56 numbers from the case with leading digit 8; and from (b), 7+6+5+4+3+2+1 = 28 numbers.  Total 56+28 = 84 numbers.&lt;br&gt;
And, finally, the total number of 4-digit numbers that meet the requirement is&lt;br&gt;
1+4+10+20+35+56+84 = 210&lt;br&gt;&lt;/pre&gt;
Like the other tutor, it seems to me there might be an easier path to the answer; but I haven&#39;t seen it yet.&lt;br&gt;
It is curious to me that all the numbers in the calculations are the C(n,r) numbers in Pascal&#39;s Triangle.  But again I haven&#39;t been able to see the logical connection between those numbers and this problem.
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