Question 1208133
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a digital timer counts down from 3 min to zero, one second at a time. 
for how many seconds does at least one of these digits show a 5?
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<pre>
(a)  First (leftmost) digit is never 5.



(b)  Consider the set of numbers XY5 from 1 to 180 inclusive with the last 
     (rightmost) digit of 5 and the middle digit Y =/=5. 

     These numbers are

           005, 015, 025, 035, 045, {{{highlight(cross(055))}}}, 065, 075, 0.85, 0.95,
           
           105, 115, 125, 135, 145, {{{highlight(cross(155))}}}, 165, 175.


     The crossed numbers are those with 5 in the middle position.

     The amount of such  "survived" numbers is 9 + 7 = 16.

     The counter counts/adds 1 as each such a number appears.



(c)  To it, we should add appearances X5Z.

     There are exactly  10 + 10 = 20  such appearances

         050, 051, 052, 053, 054, 055, 056, 057, 058, 059,

         150, 151, 152, 153, 154, 155, 156, 157, 158, 159.


     The counter adds 1 as each such a number appears.



(d)  Thus the total (the last) number on the counter at the end is the sum 16 + 20 = 36.    <U>ANSWER</U>
</pre>

Solved.


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What I did to solve the problem is THIS:


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I separated the set of numbers, containing at least one digit "5", 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in two disjoint subsets &nbsp;(b) &nbsp;and &nbsp;(c) &nbsp;in a way that the counter adds &nbsp;1 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;at every appearance of a number from each of the two subsets.