$a, b, c$ are three particular speakers among the $10$ speakers of a meeting. The number of ways of arranging all the $10$ speakers on the dais in a row so that all the three speakers $a, b, c$ do not sit together is

$A$ library has $a$ copies of one book,$b$ copies of each of two books,$c$ copies of each of three books,and single copies of $d$ books. The total number of ways in which these books can be arranged is:

If the different permutations of all the letters of the word $EXAMINATION$ are listed as in a dictionary,how many words are there in this list before the first word starting with $E$?

For $n = 1, 2, 3, . . . , 50$,let $A = \{ a_n \mid a_n = \begin{cases} (-1)^{\frac{n}{2}} (\frac{n}{2}), & \text{if } n \text{ is even} \\ (-1)^{\frac{n-1}{2}} (\frac{n-1}{2}), & \text{if } n \text{ is odd} \end{cases} \}$ and $B$ be the set of all distinct elements of $A$. The number of permutations of all the elements of set $B$ such that even integers are in increasing order is:

The number of different permutations that can be formed by taking $4$ letters at a time from the letters of the word '$REPETITION$' is

The number of $4$ digit even numbers that can be formed using $0, 1, 2, 3, 4, 5, 6$ without repetition is