What is the probability that the sum of 6 four-sided dice is less than or equal to 14? The solution given is shown below.
My question is how did they count the numerator like that?
What is the explanation for it please?
$$\begin{align}
\frac{C_6^{14}-C_1^6\times C_6^{10}+C_2^6 \times C_6^6}{4^6}&=\frac{3003-6\times 210+15\times 1}{4^6}\\
&= \frac{1758}{4^6}\\
&= \frac{879}{2048}
\end{align}$$
I understand  the denominator namely because each of the four sided dice has four choices and six of them so, all possible outcomes will be $4^6$.
I believe that the 4-sided dice here has 1, 2, 3, 4 printed on its faces.
Any help is appreciated.

The following solution uses a generating function.  Readers not familiar with generating functions can find several resources in the answers to this question: How can I learn about generating functions?
There are $4^6$ possible outcomes when rolling six four-sided dice, all of which we assume are equally likely. We want to count the number of arrangements in which the sum is less than or equal to $14$.  This is the number of solutions in integers to
$$x_1+x_2+x_3+x_4+x_5+x_6 \le 14$$
sunbject to $1 \le x_i \le 4$ for $i=1,2,3,4,5,6$.  Equivalently, we want the number of integer solutions to
$$x_1+x_2+x_3+x_4+x_5+x_6+x_7 = 14 \tag{1}$$
where $0 \le x_7$. The generating function for the number of solutions is
$$\begin{align}
f(x) &= (x+x^2+x^3+x^4)^6 \cdot (1+x+x^2+ \dots) \\
&= x^6 \left( \frac{1-x^4}{1-x} \right)^6 \cdot \frac{1}{1-x} \tag{2} \\
&= x^6 \; (1-x^4)^6 \; (1-x)^{-7} \\
&= x^6 \; (1 -6x^4+15x^8+O(x^{12})) \; \sum_{i=0}^{\infty} \binom{7+i-1}{i} \tag{3} x^i
\end{align}$$
From $(3)$ we can see that the coefficient of $x^{14}$ in $f(x)$ is
$$[x^{14}]f(x) = \binom{7+8-1}{8} - 6 \binom{7+4-1}{4} + 15 = 1758$$
so the number of solutions to $(1)$ is $1758$, and the probability of rolling a sum of $14$ or less is
$$\frac{1758}{4^6}$$
Notes:
$(2)$  Summing geometric series (twice).
$(3)$  Applying the Binomial Theorem, first for a positive exponent and then for a negative exponent.
