Let the smallest value of $k \in N$, for which the coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + \dots + (1+x)^{99} + (1+kx)^{100}, x \neq 0$, is $(43n + \frac{101}{4}) ({}^{100}C_3)$ for some $n \in N$, be $p$. Then the value of $p+n$ is:

Let $(1 + x)(1 + x + x^2)(1 + x + x^2 + x^3) \dots (1 + x + x^2 + \dots + x^{30}) = a_0 + a_1x + a_2x^2 + \dots + a_{465}x^{465}$. Then the sum $a_0 + a_2 + a_4 + \dots$ is equal to:

$\sum_{\substack{i, j=0 \\ i \neq j}}^{n} {}^{n}C_{i} {}^{n}C_{j}$ is equal to

If $(1 - x + 2x^2)^n = a_0 + a_1x + a_2x^2 + \dots + a_{2n}x^{2n}$,where $n \in N$,$x \in R$,and $a_0, a_1, a_2$ are in Arithmetic Progression $(A.P.)$,then there exists:

For non-negative integers $s$ and $r$,let $\binom{s}{r} = \begin{cases} \frac{s!}{r!(s-r)!} & \text{if } r \leq s \\ 0 & \text{if } r > s \end{cases}$. For positive integers $m$ and $n$,let $g(m, n) = \sum_{p=0}^{m+n} \frac{f(m, n, p)}{\binom{n+p}{p}}$,where for any non-negative integer $p$,$f(m, n, p) = \sum_{i=0}^{p} \binom{m}{i} \binom{n+i}{p} \binom{p+n}{p-i}$. Then which of the following statements is/are $TRUE$?
$(A)$ $g(m, n) = g(n, m)$ for all positive integers $m, n$
$(B)$ $g(m, n+1) = g(m+1, n)$ for all positive integers $m, n$
$(C)$ $g(2m, 2n) = 2g(m, n)$ for all positive integers $m, n$
$(D)$ $g(2m, 2n) = (g(m, n))^2$ for all positive integers $m, n$

$1+{ }^{n} C_{1} \cos \theta+{ }^{n} C_{2} \cos 2 \theta+\ldots+{ }^{n} C_{n} \cos n \theta$ equals