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.

if B <= D, it has B solutions {(1,B-1,D-B+1),(2,B-2,D-B+2),...}

if B>D, it has D+1 solutions

1) Cases when B <= D: (1,1) ,(1,2), ...(1,9),(2,2),....,(9,9)

No. of solutions = 1*9+2*8+3*7+....+9*1

2) Cases when B>D: (1,0),(2,0),..(9,0),(2,1),.....,(9,8)

No. of solutions = 1*9+2*8+3*7+...+8*2+9*1

Total number of solutions = 2*(9+16+21+24+25+24+21+16+9)

= 2*(165) = 330

probability = 330/90000 = 11/3000

Edited on July 15, 2010, 2:21 pm

Posted by Praneeth on 2010-07-15 14:21:14