If I roll a dice 4 times then what is the probability of getting 20 total dots? So, here the total evets = $6^4$
If I get $a,b,c,d$ doing each rolling then, $$a+b+c+d=20$$
$$where\ a,b,c,d\in\mathbb{N} \ and\ 1\le a,b,c,d\le6$$
Now, how can I solve this by stars and bars method? Or there are other easier ways to solve this?

Using generating functions: The number of ways to roll a total of $20$ on four dice is $$\begin{align}[x^{20}](x+x^2+x^3+x^4+x^5+x^6)^4 &= [x^{16}](1+x+x^2+x^3+x^4+x^5)^4 \\ &= [x^{16}]\left({1-x^6 \over 1-x}\right)^4 \\ 
&= [x^{16}]{\binom40-\binom41x^6+\binom42x^{12}-\cdots \over (1-x)^4} \\
&= [x^{16}]{\binom40 \over (1-x)^4}-[x^{10}]{\binom41 \over (1-x)^4}+[x^4]{\binom42 \over (1-x)^4} \\
&= \binom40\binom{19}3-\binom41\binom{13}3+\binom42\binom{7}3 \\ &= 35. \end{align}$$ The probability is therefore $\frac{35}{6^4} = \frac{35}{1296} \approx 2.7\%$.
