The coefficient of $$x^{70}$$ in $$x^2(1+x)^{98} + x^3(1+x)^{97} + x^4(1+x)^{96} + \ldots + x^{54}(1+x)^{46}$$ is $$^{99}C_p - ^{46}C_q$$. Then a possible value of $$p + q$$ is :

General term:
$$x^k(1+x)^{100-k},\quad k=2\text{ to }54$$

$$For(x^{70}):pick(x^{70-k}) from ((1+x)^{100-k})$$

Coefficient:
$$\binom{100-k}{70-k}$$

Sum:
$$\sum_{k=2}^{54}\binom{100-k}{70-k}$$

Let (r=70-k):
$$r=68\to16$$
$$\Rightarrow\sum_{r=16}^{68}\binom{30+r}{r}$$

Use identity:
$$\sum_{r=a}^b\binom{n+r}{r}$$
$$=\binom{n+b+1}{b}-\binom{n+a}{a-1}$$

$$=\binom{99}{68}-\binom{46}{15}$$

$$=\binom{99}{p}-\binom{46}{q}$$

$$\Rightarrow p=68,\ q=15p+q=83$$