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Dart Data

Date: 02/29/2012 at 23:00:29
From: Marne & Iliana
Subject: How many possible options with 4 chances & 3 choices?

Picture a dartboard where hitting the outer ring scores 1 point, the next
ring in scores 5, and the bullseye scores 25.

If you have four darts and you hit the dartboard with all four of them
every time, how many different scores can you make, and what are they?

I'm trying to help my 5th grade daughter with this homework problem. The
only way we can think of is to come up with all the possible combinations
and then add up the scores. So we tried to make a graph with first dart,
second dart, etc., across the top; 1 point, 5 points, 25 points down the
side; and then started making hash marks for every possibility.

Here are some possible scores we came up with:

         4        16        32        56
         8        28        36        76
        12        52        20        ...

But we are getting confused with which ones we've already tried, and it's
taking forever. There must be an easier way that we're just not getting,
such as using an equation.

Please help, or we could be counting up hash marks all night long!

Thank you!!

Date: 03/02/2012 at 14:29:49
From: Doctor Peterson
Subject: Re: How many possible options with 4 chances & 3 choices?

Hi, Marne.

Here's another way to make an orderly list: Rather than list where each
individual dart lands, list how many land in each ring.

The list can start by putting as many as possible in one ring, then
reducing that, and at each step listing all possible ways to put the rest
in the other rings.

Here's a start:

    25    5    1  total
   ---- ---- ---- -----
     4    0    0   100
     3    1    0    80
     3    0    1    76
     2    2    0    60
     2    1    1    56
     2    0    2    52

Can you finish? After getting as far as I did, I started seeing some
patterns that make it pretty easy to continue.

The idea here -- looking at the problem from a different perspective,
turning it sort of inside-out -- is very useful when a problem seems too

- Doctor Peterson, The Math Forum 

Date: 03/02/2012 at 14:46:39
From: Marne & Iliana
Subject: Thank you (How many possible options with 4 chances & 3 choices?)

That's much better than our way -- thank you!!
Associated Topics:
Middle School Probability
Middle School Word Problems

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