I re-read the solution that you posted and realized where my thinking
was going wrong. Sorry (again!) for being a thicko.
cheers,
Rolf Turner
On 13/01/2010, at 9:19 AM, Greg Snow wrote:
How trivial is probably subjective, I don't think it is much above
trivial. I would not have been surprised to see this question on
an exam in my undergraduate (300 or junior level) probability
course (the hard part was remembering the details from that class
from over 20 years ago). My favorite test question of all time
came from that course: "You have a deck of poker cards with the 3's
removed (and jokers), you deal yourself 5 cards at random, what is
the probability of getting a straight (not including straight
flushes)?"
This problem is simpler. Just think of the 8 places in the number
as urns, and the 17 1's as balls to be put into the urns. One ball
has to go in the first urn, so you have 16 left, there are choose(16
+8-1,8-1) ways to distribute 16 undistinguishable balls among 8
distinguishable urns. But that includes some solutions with more
than 9 balls in an urn which violates the digits restriction, so
subtract off the illegal counts. If we place 10 balls in the first
urn, then we have 7 remaining balls to distribute between the 8
urns or choose( 7+8-1, 7), If we place 1 ball in the first urn and
10 balls in one of the 7 other urns (7*), then there are choose( 6
+8-1, 7 ) ways to distribute the remaining 6 balls in the 8 urns.
Not too complicated once you remember (or look up) the formula for
urns and balls.
--
Gregory (Greg) L. Snow Ph.D.
Statistical Data Center
Intermountain Healthcare
greg.s...@imail.org
801.408.8111
-----Original Message-----
From: baptiste auguie [mailto:baptiste.aug...@googlemail.com]
Sent: Tuesday, January 12, 2010 12:20 PM
To: Greg Snow
Cc: r-help
Subject: Re: [R] expand.grid game
Nice --- am I missing something or was this closed form solution not
entirely trivial to find?
I ought to compile the various clever solutions given in this thread
someday, it's fascinating!
Thanks,
baptiste
2010/1/12 Greg Snow <greg.s...@imail.org>:
This also has a closed form solution:
choose(16+8-1,7) - choose(7+8-1, 7) - 7*choose(6+8-1,7)
[1] 229713
--
Gregory (Greg) L. Snow Ph.D.
Statistical Data Center
Intermountain Healthcare
greg.s...@imail.org
801.408.8111
-----Original Message-----
From: r-help-boun...@r-project.org [mailto:r-help-boun...@r-
project.org] On Behalf Of Brian Diggs
Sent: Thursday, December 31, 2009 3:08 PM
To: baptiste auguie; David Winsemius
Cc: r-help
Subject: Re: [R] expand.grid game
baptiste auguie wrote:
2009/12/19 David Winsemius <dwinsem...@comcast.net>:
On Dec 19, 2009, at 9:06 AM, baptiste auguie wrote:
Dear list,
In a little numbers game, I've hit a performance snag and I'm
not
sure
how to code this in C.
The game is the following: how many 8-digit numbers have the sum
of
their digits equal to 17?
And are you considering the number "00000089" to be in the
acceptable set?
Or is the range of possible numbers in 10000079:98000000 ?
The latter, the first digit should not be 0. But if you have an
interesting solution for the other case, let me know anyway.
I should also stress that this is only for entertainment and
curiosity's sake.
baptiste
I realize I'm late coming to this, but I was reading it in my post-
vacation catch-up and it sounded interesting so I thought I'd give
it a
shot.
After coding a couple of solutions that were exponential in time
(for
the number of digits), I rearranged things and came up with
something
that is linear in time (for the number of digits) and gives the
count
of numbers for all sums at once:
library(plyr)
nsum3 <- function(digits) {
digits <- as.integer(digits)[[1L]]
if (digits==1) {
rep(1,9)
} else {
dm1 <- nsum3(digits-1)
Reduce("+", llply(0:9, function(x) {c(rep(0,x),dm1,rep(0,9-
x))}))
}
}
nsums <- llply(1:8, nsum3)
nsums[[5]][17]
# [1] 3675
nsums[[8]][17]
# [1] 229713
The whole thing runs in well under a second on my machine (a
several
years old dual core Windows machine). In the results of nsum3, the
i-
th element is the number of "numbers" whose digits sum to i. The
basic
idea is recursion on the number of digits; if n_{t,d} is the number
of
d-digit "numbers" that sum to t, then n_{t,d} = \sum_{i\in(0,9)}
n_{t-
i,d-1}. (Adding the digit i to each of those "numbers" makes their
sum
t and increases the digits to d). When digits==1, then 0 isn't a
valid
choice and that also implies the sum of digits can't be 0, which
fits
well with the 1 indexing of arrays.
--
Brian Diggs, Ph.D.
Senior Research Associate, Department of Surgery, Oregon Health &
Science University
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