$(2n + 1) (2n + 3) (2n + 5) \dots (4n - 1)$ is equal to :

If for some $m, n$,${ }^6 C_{m}+2({ }^6 C_{m+1})+{ }^6 C_{m+2} > { }^8 C_3$ and ${ }^{n-1} P_3 : { }^n P_4 = 1 : 8$,then ${ }^n P_{m+1} + { }^{n+1} C_m$ is equal to

Statement-$1$: The number of $4$-digit numbers that can be formed using the digits $1, 2, 3, 4, 5, 6, 7$ which are divisible by $4$ is $200$.
Statement-$2$: $A$ number is divisible by $4$ if its unit digit is divisible by $4$.

If $\sum\limits_{r = 0}^{25} {\left( {^{50}C_r \cdot ^{50 - r}C_{25 - r}} \right) = K\left( {^{50}C_{25}} \right)}$,then $K$ is equal to

Consider the following statements:
$i.$ The number of ways of placing $n$ distinct objects in $k$ distinct bins $(k \leq n)$ such that no bin is empty is ${}^{n-1}C_{k-1}$.
$ii.$ The number of ways of writing a positive integer $n$ as a sum of $k$ positive integers is ${}^{n-1}C_{k-1}$.
$iii.$ The number of ways of placing $n$ distinct objects in $k$ distinct bins such that at least one bin is non-empty is ${}^{n-1}C_{k-1}$.
$iv.$ ${}^nC_k - {}^{n-1}C_k = {}^{n-1}C_{k-1}$.

The remainder of $n^4-2n^3-n^2+2n-26$ when divided by $24$ is: