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Mountain Motif (Posted on 2010-04-07) Difficulty: 2 of 5
A five digit positive integer is a mountain number if the first three digits are in strictly ascending order and the last three digits are in strictly descending order. For example, 46872 is a mountain number, but none of 43434, 54456 and 46766 is a mountain number.

Determine the probability that N is a mountain number, given that N is a positive integer chosen at random between 40000 and 99999 inclusively.

No Solution Yet Submitted by K Sengupta    
Rating: 2.0000 (1 votes)

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Some Thoughts Let's generalize- no middle point needed | Comment 6 of 8 |
 

Toward globalization

 

 

              

    If we skip the middle digit demand, i.e. the number of ascending    and descending   members being not  necessarily equal, the  outcome will be much larger- but the calculation difficulty the same.

    I'll try to go for it.

     

    If 9 is the "summit" ,not  neccesserally  the middle, then from the up - set  12345678 we can choose any  subset, except the empty set , and from the down -set  876543210 we can choose any  subset,  except the empty set ,
    so there are     2^8-1)*(2^9-1)        choices.                              
     255*511=130305

    For 8 as the "summit" (2^7-1)*(2^8-1) choices     127*255=32385.                                     

     

    For 7 as  the "summit"  (2^6-1)*(2^7-1) choices      63*127=8001

    For 6 as  the "summit"   (2^5-1)*(2^6) choices.          31*63=1953

    For 5 as  the "summit"  (2^4-1)*(2^5-1) choices         15*31=465

    For4 as  the "summit"   (2^3-1)*(2^4-1) choices.           7*15=105

    For  3 as  the "summit"  (2^2-1)*(2^3-1) choices.              3*7=21

    For 2 as  the    "summit"   (2^1-1)*(2^2-1) choices.           1*3=3

     

    173,238  from 123456789876543210 to 120 inclusive.

 

Edited on April 8, 2010, 2:07 pm
  Posted by Ady TZIDON on 2010-04-08 14:02:58

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