Gavin buys a lottery ticket, and in accordance with the rules he picks six different integers from through 1 to 46 inclusively. He chooses his numbers so that the sum of the baseten logarithms of his six numbers is an integer.
By coincidence, the integers on the winning ticket have the same property  that is: the sum of the baseten logarithms of the six numbers is an integer.
What is the probability that Gavin holds the winning ticket?
There are 12 numbers that are either powers of 2, of 5 or a combination thereof between 1 and 45: 1, 2, 4, 8, 16, 32, 5, 25, 10, 20, 40, 50.
They can be multiplied in various combinations of six to produce a power of 10, and the following program finds these ways:
DefDbl AZ
Dim crlf$, num(12)
Private Sub Form_Load()
ChDir "C:\Program Files (x86)\DevStudio\VB\projects\flooble"
Text1.Text = ""
crlf$ = Chr(13) + Chr(10)
Form1.Visible = True
DoEvents
num(1) = 1
num(2) = 2
num(3) = 4
num(4) = 8
num(5) = 16
num(6) = 32
num(7) = 5
num(8) = 25
num(9) = 10
num(10) = 20
num(11) = 40
num(12) = 50
For a = 1 To 7
For b = a + 1 To 8
For c = b + 1 To 9
For d = c + 1 To 10
For e = d + 1 To 11
For f = e + 1 To 12
prod = num(a) * num(b) * num(c) * num(d) * num(e) * num(f)
If prod = 10 Or prod = 100 Or prod = 1000 Or prod = 10000 Or prod = 100000 Or prod = 1000000 Or prod = 10000000 Or prod = 100000000 Then
Text1.Text = Text1.Text & Str(num(a)) & Str(num(b)) & Str(num(c)) & Str(num(d)) & Str(num(e)) & Str(num(f))
Text1.Text = Text1.Text & " " & prod & crlf
ct = ct + 1
End If
Next
Next
Next
Next
Next
Next
Text1.Text = Text1.Text & crlf & Str(ct) & " done"
End Sub
numbers product
1 2 4 5 25 10 10000
1 2 4 25 10 50 100000
1 2 8 5 25 50 100000
1 2 5 25 10 40 100000
1 2 5 10 20 50 100000
1 2 25 10 40 50 1000000
1 4 5 25 10 20 100000
1 4 5 25 40 50 1000000
1 4 25 10 20 50 1000000
1 8 5 25 20 50 1000000
1 16 5 25 10 50 1000000
1 5 25 10 20 40 1000000
1 25 10 20 40 50 10000000
2 4 5 25 20 50 1000000
2 8 5 25 10 50 1000000
2 5 25 20 40 50 10000000
4 5 25 10 40 50 10000000
8 5 25 10 20 50 10000000
There are 18 ways, so Gavin's probability of a win is 1/18.

Posted by Charlie
on 20141006 14:19:08 