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+1 vote
How many ways, can sum be equal to 12 of 3 dice?

where 1<=x1<=6;1<=x2<=6; 1<=x3<=6
How to solve it further?
asked in Combinatory by Veteran (14.6k points)   | 184 views
Bars and stars approach can also be used, if we restrict one star each in 3 partitions we will have 11C9 combinations i.e 55 . From this we should subtract case when any partition has greater than 7 stars. 7 star in x1 , x2 1 star and x3 1 star, select remaining 3 stars from 3 star+2 bars i.e 5 C 3 .This will be 3 times as we can place 7 stars in x1 or x2 or x3. So 11C9 - 3(5C3) = 25. But I think generating functions process is much easier.

2 Answers

+6 votes
Best answer

Higher order terms are neglected and only $x^3$ and $x^9$ terms are useful.

answered by Veteran (11.4k points)  
selected by

you explained very well! much obliged!! 
Please let me know how how did you reach this equation
and why are you loking for the coefficient of x^12, please provide details

it is same like distribution of indistinguishable balls into distinguishable boxes.

Say where x1+x2+x3=12 

   1<=x1<=6;1<=x2<=6; 1<=x3<=6

x1,x2,x3 can have value between 1,2,3,4,5,6.

Each have the same (x+x^2+x^3+x^4+x^5+x^6) so taken cube ....and n=12 so finding 12th term

Can you please explain this part from your solution:

on RHS side you it is a expansion $\binom{n+2}{n}x^{n}$

I want $x^{12}$ then so did $x^3 * x^9$ and $-3x^9 * x^3$

coefficient of x^9 is 11C9 (for first ) and x^3  is 5c3.
thanking you in words not enough!! i knew nothing about generating functions. now i am much comfortable with them. Much obliged!!
Welcome  Vijay :)
–1 vote
I guess we can solve it using Generating function Concept
I am getting answer as 70
answered by Loyal (3.7k points)  
edited by
give your detaied explanation please

find coeff of x12 in (x^6+x^5+x^4+x^3+x^2+x)3

can you please tell how did you reach this equation and how to calculate coeffcient of x^12 in it?
I am not that much stronger in maths, hence a detailed solution is required
@vijay i think u should read generating function from rosen book so that u ll get every thing that u want 2 know, it is hardly takes 3-4 hours and after that if u have any doubt u can ask here.:)

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