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Evaluate probabilities (Posted on 2016-01-26) Difficulty: 2 of 5
Scanning a number from left to right, if the next digit is greater in value than the preceeding digit, we say that the number is strictly increasing: 1589, 3679, and 1348 are examples of 4-digit numbers with this property.

Given a certain number is strictly increasing, what is the probability that it contains

a. exactly 6 digits?
b. 6 digits or more?

No Solution Yet Submitted by Ady TZIDON    
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Solution Counting, Counting (spoiler) | Comment 1 of 3
Let's assume that we are only talking about integers.

Line the digits 1 through 9 up and form an increasing number by striking out 0 to 9 digits.
There are 2^9 ways of doing this, but let's also rule out 0 and 1 digit numbers.
Then the total number of valid increasing numbers are 2^9 - 10 = 512 - 10 = 502

Part a) To form a 6 digit number, strike out 3 of the 9 digits.  The number of ways of doing this is C(9,3) = 9*8*7/3*2*1 = 84.  
so the desired probability = 84/502 = 42/251, which is approximately 1/6

Part b) (84+36+9+1)/502 = 130/502 = 65/152

If we allow non-integers, then the probability tilts in favor of larger numbers, because 12 can be 12 or 1.2 or .12 (three possibilities) whereas a 6 digit number has seven possibilities.  Since this is D2, I guess that this is not what the author had in mind.

  Posted by Steve Herman on 2016-01-26 09:15:13
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