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 complicated. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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!!
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