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Hill Numbers Settlement (Posted on 2010-07-15) Difficulty: 2 of 5
A 5-digit base ten positive integer of the form ABCDE is called a hill number if the digits B and D are each equal to the sum of the digits to their immediate left and right, that is, B = A + C and D = C + E. (Each of the capital letters in bold denotes a digit from 0 to 9, whether same or different.)

Determine the probability that x is a hill number, given that x is a base ten positive integer chosen at random between 10000 and 99999 inclusively.

No Solution Yet Submitted by K Sengupta    
Rating: 1.6667 (3 votes)

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Solution By counting (spoiler) | Comment 3 of 4 |
If C = 0, DE can have 10 (00,11,..99) values and AB can have 9 (11,22,..99).  Since they are independent, there are 10*9 = 90 hill numbers where C = 0.

Similarly, if C = 1, DE can have 9 (10,21,..,98) values and AB can have 8 (12,23,..89). Since they are independent, there are 9*8 = 72 hill numbers where C = 1.

Similarly, there are 8*7 = 56 hill numbers where C = 2.

Total hill numbers = 10*9 + 9*8 + 8*7 + ... + 2*1 + 1*0 = 330.

Probability = 330/90000 = 11/3000.

Same answer as Ed.

  Posted by Steve Herman on 2010-07-15 14:27:10
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