# A die is thrown five times, what is the probability that you get 20 as the sum of the values

This is supposed to be a Inclusion-Exclusion problem.

We have $6^5=7776$ different results.

Now, with the Inclusion-Exclusion principle i resolve the number of solutions for the equation:

$d_1+d_2+d_3+d_4+d_5=20 , 1 \leq d_i \leq 6 ,\forall 1 \leq i \leq 5$

That, i think, is equivalent to resolve:

$d_1+d_2+d_3+d_4+d_5=15 , 0 \leq d_i \leq 5 ,\forall 1 \leq i \leq 5$

This is:

$\binom{19}{15}-\binom{5}{1} \binom{13}{9}+\binom{5}{2} \binom{7}{3}=651$

So mi answer is: $\frac{651}{7776}$.

However, the "correct answer" is: $\frac{116}{7776}$. Whats the problem with mi reasoning?

EDIT I found the solutions for my problem in the book, the result is: $\frac{651}{7776}$.

I was right, but the solution from mi teacher was wrong. Thats why the confusion.

• You probably mean $6^5$ different results? Feb 11 '14 at 18:28
• @gt6989b Yes, my mistake. Feb 11 '14 at 18:28
• How do you get that expression that evaluates to $651$? Feb 11 '14 at 18:32
• @ShreevatsaR Trying to simplifiy the original equation. I assign one element to the five variables, an then decrease the right side: $20-5=15$ Feb 11 '14 at 18:35
• @ShreevatsaR wolframalpha.com/input/…*+%5Cbinom%7B13%7D%7B9%7D%2B%5Cbinom%7B5%7D%7B2%7D+*%5Cbinom%7B7%7D%7B3%7D shows this evaluates to 651 Feb 11 '14 at 18:46

Using generating functions, we get a generating function of $$f(x) = 1+x+\ldots+x^5 = \frac{1-x^6}{1-x}$$ for each variable so we are now looking for the coefficient of $x^{15}$ in $f(x)^5$: $$\begin{split} \left[x^{15}\right]\frac{\left(1-x^6\right)^5}{(1-x)^5} &= \left[x^{15}\right]\frac{1-5x^6+10x^{12}}{(1-x)^5} \\ &= \left[ \left[x^{15}\right] -5\left[x^{9}\right] + 10 \left[x^{3}\right]\right] \frac{1}{(1-x)^5}\\ &= 3876 - 5 \cdot 715 + 10 \cdot 35 \\ &= 651. \end{split}$$
• $336$ would be the numerator for your probability fraction. Feb 11 '14 at 18:47
• @Wyvern666 I'm not making any claim as to the accuracy of the method. I'm just answering your question about what the $336$ means. If I could answer accurately myself, I would. I leave it to those with more knowledge. Feb 11 '14 at 19:03
• @vonbrand actually checked the expansion, your coefficients come out the same, $\binom{19}{4} = 3876, \binom{13}{4} = 715, \binom{7}{4} = 35$. Wolfram Alpha doesn't lie :) Feb 11 '14 at 19:56
You can calculate this by finding the coefficient on $x^{20}$ in the polynomial $$\left(\frac{1}{6}x+\frac{1}{6}x^2+\frac{1}{6}x^3+\frac{1}{6}x^4+\frac{1}{6}x^5+\frac{1}{6}x^6 \right)^5$$ Using PARI/GP, I get that this polynomial is $$\frac{1}{7776} x^{30} + \frac{5}{7776} x^{29} + \frac{5}{2592} x^{28} + \frac{35}{7776} x^{27} + \frac{35}{3888} x^{26} + \frac{7}{432} x^{25} + \frac{205}{7776} x^{24} + \frac{305}{7776} x^{23} + \frac{35}{648} x^{22} + \frac{5}{72} x^{21} + \frac{217}{2592} x^{20} + \frac{245}{2592} x^{19} + \frac{65}{648} x^{18} + \frac{65}{648} x^{17} + \frac{245}{2592} x^{16} + \frac{217}{2592} x^{15} + \frac{5}{72} x^{14} + \frac{35}{648} x^{13} + \frac{305}{7776} x^{12} + \frac{205}{7776} x^{11} + \frac{7}{432} x^{10} + \frac{35}{3888} x^9 + \frac{35}{7776} x^8 + \frac{5}{2592} x^7 + \frac{5}{7776} x^6 + \frac{1}{7776} x^5$$ This shows that your value, $\frac{651}{7776}=\frac{217}{2592}$ is correct.