PDF of 6 six-sided fair dice sum I am trying to create a pdf for the sum of 6 six-sided fair dice. The sums range from 6 to 36 (either have all 1s or all 6s).

My attempt at the solution:

$x$ = sum of the 6 dice

$P(x=6) = C(6,0) (1/6)^6$  &lt;--- all 1s

$P(x=7) = C(6,1) (1/6)^5 (1/6)$ &lt;--- five 1s and one 2

$P(x=8) = C(6,1) (1/6)^5 (1/6) + C(6,2) (1/6)^4 (1/6)^2$  &lt;----- five 1s and one 3 or four 1s and two 2s

$P(x=9) = C(6,1) (1/6)^5 (1/6) + 2\cdot C(6,2) (1/6)^4 (1/6) (1/6)$ &lt;----- five 1s and one 4 or four 1s, one 3 and one 2

$P(x=10) = C(6,1) (1/6)^5 (1/6) + 2\cdot C(6,2) (1/6)^4 (1/6) (1/6) + C(6,2) (1/6)^4 (1/6)^2$ &lt;----- five 1s and one 6 or four 1s, one 4 and one 2 or four 1s and two 3s

As you can see this process is extremely tedious to count all the possible combinations as the sums get larger, so I was wondering if anyone has a more concise approach/solution to this particular problem.

Thanks!
You are looking for the coefficients of the expansion of $$\left(\dfrac{x+{x}^{2}+{x}^{3}+{x}^{4}+{x}^{5}+{x}^{6}}{6}\right)^6=\left(\dfrac{x(1-x^6)}{6(1-x)}\right)^6.$$  This will give $$\frac{{x}^{6}}{46656}+\frac{{x}^{7}}{7776}+\frac{7\,{x}^{8}}{15552}+\frac{7\,{x}^{9}}{5832}+\frac{7\,{x}^{10}}{2592}+\frac{7\,{x}^{11}}{1296}+\frac{19\,{x}^{12}}{1944}+\frac{7\,{x}^{13}}{432}+\frac{43\,{x}^{14}}{1728}+\frac{833\,{x}^{15}}{23328}+\frac{749\,{x}^{16}}{15552}+\frac{119\,{x}^{17}}{1944}+\frac{3431\,{x}^{18}}{46656}+\frac{217\,{x}^{19}}{2592}+\frac{469\,{x}^{20}}{5184}+\frac{361\,{x}^{21}}{3888}+\frac{469\,{x}^{22}}{5184}+\frac{217\,{x}^{23}}{2592}+\frac{3431\,{x}^{24}}{46656}+\frac{119\,{x}^{25}}{1944}+\frac{749\,{x}^{26}}{15552}+\frac{833\,{x}^{27}}{23328}+\frac{43\,{x}^{28}}{1728}+\frac{7\,{x}^{29}}{432}+\frac{19\,{x}^{30}}{1944}+\frac{7\,{x}^{31}}{1296}+\frac{7\,{x}^{32}}{2592}+\frac{7\,{x}^{33}}{5832}+\frac{7\,{x}^{34}}{15552}+\frac{{x}^{35}}{7776}+\frac{{x}^{36}}{46656}.$$
Yes considering that there are 6^6 = 46,656 combinations, counting them one-by-one isn't very practical. There is some good info here:

http://mathworld.wolfram.com/Dice.html