Re: Possible results from three variables


"Robert Israel" <israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> skrev i melding
news:rbisrael.20070712220809$4d31@xxxxxxxxxxxxxxxxxxx
"Karl-Olav Nyberg" <konyberg@xxxxxxxxx> writes:


<bakpao@xxxxxxxxx> skrev i melding
news:1184225137.530634.258440@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Given three variables, say x, y, z with each variable being an integer
from 1 to 10, how many possible values are there from the equation of
x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
This is wrong because some of the combinations actually the same, e.g.
1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.

Is there any formula that we can use to find out the number of
possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?

In Maple:

a:= n -> nops({seq(seq(seq(x*y*z,x=1..n),y=1..n),z=1..n)});


Hi.

(1*2*3*4*5*6*7*8*9*10) / (1*2*3*1*2*3*4*5*6*7) = 120

Generally: (n!) / (m! * (n-m)!)

It's a coincidence that (10 choose 3) is the right answer. For integers
1 to 11 the answer would be 173, not (11 choose 3) = 165.

See sequence A027425 in the On-Line Encyclopedia of Integer Sequences,
<http://www.research.att.com/~njas/sequences/A027425>. I don't think
there's a closed form for a(n) or a closed form generating function or
recurrence.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada V6T 1Z2

Hi.
I looked up this sequence earlier today (I have On-Line E... as a favourite
in IE). And I admit I was to fast on the trigger. It doesn't look like this
problem has an easy answer. I see they are using A027425 to generate A027426
and visa versa. Looking at quasi's replays there are some amazing things
going on. There are clear relations between some of the f(n,k) and for
others "none".

A very fun problem.

Karl-Olav Nyberg


.

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