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# We have 10 standard 6-sided dice, all different colors. In how many ways can we roll them to get a sum of 20?

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We have 10 standard 6-sided dice, all different colors. In how many ways can we roll them to get a sum of 20?

Please explain!

@Melody, if I solve the answer before you do, I will remember to get on here and check your answer! I'm sorry you think I'm rude.

Oct 13, 2017
edited by Jeff123  Oct 13, 2017

### 7+0 Answers

#1
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There are 85,228 ways of getting 20 when rolling 10 dice. The probability of rolling a total of 20 is:

85,228 / (6^10) =85,228 / 60,466,176 =0.00140951 x 100=0.140951%.

The calculations for these odds are exactly the same as calculating the coefficient of x^20 in the expansion of this: (x+x^2+x^3+x^4+x^5+x^6)^10. So that x^20 =85,228x^20.

To read about this in detail, see this page of "Mathworld": http://mathworld.wolfram.com/Dice.html

Oct 13, 2017
#2
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I'm not sure I understand the website. Can you rephrase it so it is simpler?

I found out that we must find how many ways are there to give out 20 dots to 10 dice. Try simplifying that and find a pattern... I'm working on that right now

Oct 13, 2017
#3
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CPhill can explain it much better than I can. He can show you how to calculate the coefficient of x^20 in the expansion of the above sequence.

Oct 13, 2017
#4
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Oh okay, but thanks so much!!!

Oct 13, 2017
#5
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You still have not responded to my last answer even though I specifically requested you to.

That was 2.5 weeks ago.  Why would I bother responding to this new question?

Oct 13, 2017
#6
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Here is your question posted here and the answers. Go through it in detail:

http://web2.0calc.com/questions/counting-question#r17

Oct 13, 2017
#7
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This is the one I was actually referring to:

https://web2.0calc.com/questions/probability-and-geometry#r7

Melody  Oct 13, 2017