Determine the total number of 8-digit base ten positive integers such that the sum of the four leftmost digits is equal to precisely 15 more than the sum of four rightmost digits.
*** Assume non leading zero for each of the numbers.
The first four digits must not allow leading zeros, but any digit including zero may be in the 5th position, so the last four digits may begin with zero.
Ways for given s.o.d. among 4 digits
sum of digits 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
w/o leading zeros 0 1 4 10 20 35 56 84 120 165 219 279 342 405 465 519 564 597
with leading zeros 1 4 10 20 35 56 84 120 165 220 282 348 415 480 540 592 633 660
18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
615 615 597 564 519 465 405 342 279 219 165 120 84 56 35 20 10 4 1
670 660 633 592 540 480 415 348 282 220 165 120 84 56 35 20 10 4 1
The ways for a full 8-digit number is the number of ways for each part separately, multiplied together:
1 * 519 = 519
4 * 564 = 2256
10 * 597 = 5970
20 * 615 = 12300
35 * 615 = 21525
56 * 597 = 33432
84 * 564 = 47376
120 * 519 = 62280
165 * 465 = 76725
220 * 405 = 89100
282 * 342 = 96444
348 * 279 = 97092
415 * 219 = 90885
480 * 165 = 79200
540 * 120 = 64800
592 * 84 = 49728
633 * 56 = 35448
660 * 35 = 23100
670 * 20 = 13400
660 * 10 = 6600
633 * 4 = 2532
592 * 1 = 592
total 911304 such 8-digit integers.
DefDbl A-Z
Dim crlf$, nlzero(36), lzero(36)
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
For a = 0 To 9
For b = 0 To 9
For c = 0 To 9
For d = 0 To 9
t = a + b + c + d
If a <> 0 Then nlzero(t) = nlzero(t) + 1
lzero(t) = lzero(t) + 1
Next
Next
Next
Next
For i = 0 To 36
Text1.Text = Text1.Text & mform(i, "###0")
Next
Text1.Text = Text1.Text & crlf
For i = 0 To 36
Text1.Text = Text1.Text & mform(nlzero(i), "###0")
Next
Text1.Text = Text1.Text & crlf
For i = 0 To 36
Text1.Text = Text1.Text & mform(lzero(i), "###0")
Next
Text1.Text = Text1.Text & crlf
For low = 0 To 36 - 15
Text1.Text = Text1.Text & mform(lzero(low), "###0") & " *" & mform(nlzero(low + 15), "###0") & " =" & mform(lzero(low) * nlzero(low + 15), "#####0") & crlf
totnumber = totnumber + lzero(low) * nlzero(low + 15)
Next
Text1.Text = Text1.Text & totnumber & crlf & " done"
End Sub
Function mform$(x, t$)
a$ = Format$(x, t$)
If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
mform$ = a$
End Function
|
|
Posted by Charlie
on 2015-09-28 15:56:47 |
the computer doesn't tire of writing the numbers
