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# Sticks and Stones Problem (I am kind of stumped...)

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Use sticks and stones to solve this :

Uncle Henry is feeling generous and has decided to distribute four \$1 bills, three \$5 bills, two \$20 bills, and a \$100 bill to three nephews. How many different ways can he distribute the money?

Are the nephews, in this case, the stones? And the money the sticks?

How should I set up my combination?

May 12, 2019
edited by CalculatorUser  May 12, 2019

#1
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Here is my naive approach:
Subsets [O, O, O, O, F, F, F, T, T, H, {3} ] =53
O=\$1, F=\$5, T=\$20 and H=\$100
{F, F, F} | {F, F, H} | {F, F, O} | {F, F, T} | {F, H, F} | {F, H, O} | {F, H, T} | {F, O, F} | {F, O, H} | {F, O, O} | {F, O, T} | {F, T, F} | {F, T, H} | {F, T, O} | {F, T, T} | {H, F, F} | {H, F, O} | {H, F, T} | {H, O, F} | {H, O, O} | {H, O, T} | {H, T, F} | {H, T, O} | {H, T, T} | {O, F, F} | {O, F, H} | {O, F, O} | {O, F, T} | {O, H, F} | {O, H, O} | {O, H, T} | {O, O, F} | {O, O, H} | {O, O, O} | {O, O, T} | {O, T, F} | {O, T, H} | {O, T, O} | {O, T, T} | {T, F, F} | {T, F, H} | {T, F, O} | {T, F, T} | {T, H, F} | {T, H, O} | {T, H, T} | {T, O, F} | {T, O, H} | {T, O, O} | {T, O, T} | {T, T, F} | {T, T, H} | {T, T, O}

May 13, 2019
#2
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Um I had an entirely different answer (probably not correct I will have to see what the answer is)

Using stars and bars

| | | Which represents nephews

O O O O F F F T T H Which represents money

So we have | | | O O O O F F F T T H

Ok so I think it is permutation.

So permutation formula

13! / (13-10)!

1716 ways???

Please, if someone is good at this, explain and check our answers.

May 13, 2019
#3
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We can split this up into multiple cases to consider.

One-dollar bills: Set up the dividers to get  (7-1)C(2)=6C2=15 ways.

Five-dollar bills Again...use the same method to get (6-1)C2=5C2=10 ways.

Twenty-dollar bills: (5-1)C2=6 ways.

Hundred-dollar bills: Any three nephews can get it, so there are three ways or (4-1)C2=3C2=3 ways.

Thus, the answer is 2700 ways.

May 13, 2019
#4
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Ohhh, ok, I totally screwed up. Thanks Tertre!

CalculatorUser  May 13, 2019
#5
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No problem...this is a very hard concept to grasp. I could give you a few more practice problems if you would like.

tertre  May 13, 2019
#6
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Yes, please, I need to understand this.

CalculatorUser  May 13, 2019
#7
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1. How many positive solutions are there to w+x+y+z=60?

2. 6 students are running for class president out of 20 members. How many vote counts are possible, if some people decide not to vote?

After you finish these problems, I will give you new ones.

Enjoy!

May 13, 2019
#8
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Look at this somewhat similar problem on the Internet: https://www.rainbowresource.com/pdfs/products/prod060689_smp02.pdf

May 13, 2019