$3$ boys $B_i, i = 1, 2, 3$ and $6$ girls $G_i, i = 1, 2, . . . , 6$ are to be seated in a row. The number of ways they can be seated so that $B_1, B_2$ are separated and $G_1, G_2$ are also separated is equal to:

The sum of all possible numbers that can be formed by using the digits $2, 3, 5, 7$ without repetition of digits is

For a natural number $n$,the inequality $2^n(n - 1)! < n^n$ holds true if:

Consider the following statements regarding $n$ identical objects and $k$ distinct bins:
$i.$ The number of ways of placing $n$ identical 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 $n$ identical objects in $k$ distinct bins such that at least one bin is non-empty is ${}^{n-1}C_{k-1}$.
$iv.$ ${}^nC_k - {}^{n-1}C_k = {}^{n-1}C_{k-1}$.
Which of the above statements are true?

$A$ five-digit number divisible by $3$ has to be formed using the numerals $0, 1, 2, 3, 4,$ and $5$ without repetition. The total number of ways in which this can be done is:

How many words can be formed using $3$ consonants and $2$ vowels from $5$ consonants and $4$ vowels?