All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Probability
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 2 of 8 |

The peak (middle digit) of the mountain must be at least 6 as the first digit must be 4 or higher. The first portion of the number can start at 4, 5, 6 or 7. The number of possible first digits depends on the choice of the second digit:

peak      second digit
digit     8   7   6   5     sum
-----    --- --- --- ---    ---
  9       4   3   2   1      10
  8           3   2   1       6
  7               2   1       3
  6                   1       1

As an example, when the second digit is a 7, there are 3 ways of choosing the first digit: 6, 5 or 4, so that 3 is the number that appears on the table above, under 7.

The last digit does not have the same restriction as the first (that is, needing to be 4 or higher), so the various fourth digits include more possibilities:

peak          fourth digit
digit       8   7   6   5   4   3   2   1         sum
-----      --- --- --- --- --- --- --- ---        ---
  9         8   7   6   5   4   3   2   1          36
  8             7   6   5   4   3   2   1          28
  7                 6   5   4   3   2   1          21
  6                     5   4   3   2   1          15

For example, when the fourth digit is 5, the last digit can be anything from 0 to 4, which is five possibilities.

For each of the four possible peak digits, you can combine any of the first/second digit pairs with any of the fourth/fifth digit pairs, so the total ways of producing a mountain number is:

10*36 + 6*28 + 3*21 + 1*15 = 606

This is out of a total of 99999 - 40000 + 1 = 60000, making the probability 606/60000 = 101/10000 = 0.0101.

Edited on April 7, 2010, 11:46 am
  Posted by Charlie on 2010-04-07 11:44:17

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (0)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information