# How many numbers are straights?

As many people, I have to deal with a lot of numbers all the time (bug numbers, changelist numbers, tracking numbers, ...) and as a mathematician I sometimes notice when numbers have peculiar properties.

For example, I recently ran into 152463, which is a straight (made up of consecutive digits, not necessarily in order.)  I then ran into 35426, which is also a straight.  Is this odd?

To put it another way - what proportion of numbers are straights?

We consider only non-negative integer numbers {0, 1, 2, ... }.  No duplicate digits are allowed, so the very last straight is 9,876,543,210.

Break it down by the length of the number in digits.  For example, how many four-digit numbers are there?  We can choose any number from 1-9 for the first digit, and any number from 0-9 for each of the other three digits; so there are 9 * 103 numbers of length 4.

How many of these are straights?  There are seven ways to choose a set of four consecutive digits to make up a four-digit straight: {0,1, 2, 3}, {1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6}, {4, 5, 6, 7}, {5, 6, 7, 8}, {6, 7, 8, 9}.

Each of these seven ways can then be shuffled to make the straights themselves.  For six of these seven sets of digits, all possible shufflings are allowed; there are 4! shufflings for each set.  This gives 6 * 4! straights.

But for {0, 1, 2, 3}, we have to be careful not to end up with 0 as a leading digit.  We choose the leading digit first out of the set {1, 2, 3} (there are three ways to do this) and then we have complete freedom to shuffle the three remaining digits (including 0.)  This gives 3 * 3! straights.

So there are 6 * 4! + 3 * 3! straights which are four digits long.

Considering all possible lengths from 1 to 10 (we'll consider 0 as a number of length 1) gives the following table:

 Length Numbers Straights % Straights 1 10 10 100.000% 2 9 * 101 17 18.889% 3 9 * 102 46 5.111% 4 9 * 103 162 1.800% 5 9 * 104 696 0.773% 6 9 * 105 3480 0.387% 7 9 * 106 19,440 0.216% 8 9 * 107 115,920 0.129% 9 9 * 108 685,440 0.076% 10 9 * 109 3,265,920 0.036% Total 1010 40,91,131 0.041%

Since roughly 1% of five-digit numbers are straights, it is not surprising that I see them that often.

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