6. The coefficient of $x^{150}$ in the expansion of $\left(1+x+x^{2}+\cdots+x^{100}\right)^{3}$, after combining like terms, is $\qquad$ (answer with a number).
6. 7651 .

By the polynomial multiplication rule, the problem can be transformed into finding the number of non-negative integer solutions to the equation
$$
s+t+r=150
$$

that do not exceed 100. Clearly, the number of non-negative integer solutions to equation (1) is $\mathrm{C}_{152}^{2}$.
Next, we find the number of non-negative integer solutions to equation (1) where one variable exceeds 100.
Since their sum is 150, only one number can exceed 100. Without loss of generality, let $s>100$. Transform equation (1) into
$$
(s-100)+t+r=49
$$

Let $s^{\prime}=s-100$, then the number of non-negative integer solutions to the equation $s^{\prime}+t+r=49$ is $\mathrm{C}_{51}^{2}$. Therefore, the coefficient of $x^{150}$ is $\mathrm{C}_{152}^{2}-\mathrm{C}_{3}^{1} \mathrm{C}_{51}^{2}=7651$.