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Cinq Digit Choice (Posted on 2014-09-22) Difficulty: 3 of 5
A five digit positive integer N without a leading zero is selected at random.

Determine the probability that the sum of the first digit, third digit and fifth digit of N is equal the product of its second digit and fourth digit.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution (spoiler) Comment 1 of 1
DefDbl A-Z
Dim crlf$

Function mform$(x, t$)
  a$ = Format$(x, t$)
  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
  mform$ = a$
End Function

Private Sub Form_Load()
 ChDir "C:\Program Files (x86)\DevStudio\VB\projects\flooble"
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 DoEvents
 
 For n = 10000 To 99999
  s$ = LTrim(Str(n))
  prod = Val(Mid(s$, 2, 1)) * Val(Mid(s$, 4, 1))
  Sum = Val(Mid(s$, 1, 1)) + Val(Mid(s$, 3, 1)) + Val(Mid(s$, 5, 1))
  ct = ct + 1
  If Sum = prod Then
    goodct = goodct + 1
    Text1.Text = Text1.Text & n & Str(goodct) & Str(ct) & mform(goodct / ct, "#0.0000000") & crlf
    DoEvents
  End If
 Next
 Text1.Text = Text1.Text & Str(goodct) & Str(ct) & mform(goodct / ct, "#0.0000000") & crlf
 
 Text1.Text = Text1.Text & crlf & " done"
End Sub

finds

 1738 90000 0.0193111
 
which is 1738 such numbers out of the 90000 numbers in the range for a probability of 1738/90000 = 869/45000 = 0.019311111....

An analytic method would examine the 9*8/2=36 possible products of different digits and the 9 possible products of equal digits that could result in a positive sum, and see how many ways three single-digit numbers, the first of which can't be zero, can add up to the respective products. Any products greater than 27 could be ignored of course.

Needing consideration are:

1*1
1*2
1*3
1*4
1*5
1*6
1*7
1*8
1*9

2*2
2*3
2*4
2*5
2*6
2*7
2*8
2*9

3*3
3*4
3*5
3*6
3*7
3*8
3*9

4*4
4*5
4*6

5*5

Sets of digits adding to the products of the same integer (i.e., squares) count only once, while those of unequal digits count twice, Completion is "left as an exercise for the reader".

  Posted by Charlie on 2014-09-22 16:29:18
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