Re: Combinatoric question

In article <4463FB4E.281D@xxxxxxx>, Larry Coon <lmcoon_nospam@xxxxxxx> wrote:
Not homework or anything like that -- just curiosity.
Been too long since college math classes, and I didn't
turn up anything via Google.

What's the correct approach to problems like, "find
how many n-digit numbers have the digit d at least x
times." E.g., "find how many 10-digit numbers have
at least five sevens."

The number of n-digit numbers (with leading zeros allowed)
having the digit d exactly x times may be obtained as
follows. Think of how you might select such a number.
1) choose x places out of n to put the digit d
2) fill the other n-x places with the other 9 digits.
So the number of ways to do this is (n choose x) 9^(n-x).

The number of n-digit numbers (with leading zeros allowed)
having the digit d at least x times is thus
sum_{y=x}^n (n choose y) 9^(n-y)

This can be written using a hypergeometric function, but
it's easier to just do the sum.

As was mentioned, if you don't allow leading zeros (and d
is not 0), just subtract the value for n-1 from the value
for n.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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