How many words (with or without meaning) of $3$ distinct English alphabets can be formed?

If $\sum_{i=0}^4 {^{4+i}}C_i + \sum_{j=6}^9 {^{3+j}}C_j = {^x}C_y$ ($x$ is a prime number),then which one of the following is incorrect?

If the letters of the word $MOTHER$ are permuted and all the words so formed (with or without meaning) are listed as in a dictionary,then the position of the word $MOTHER$ is

In how many different ways can the letters of the word '$OFFICES$' be arranged?

$A$ dictionary is printed consisting of $7$-letter words only that can be formed using the letters of the word $CRICKET$. If the words are arranged in alphabetical order,as in an ordinary dictionary,then the number of words before the word $CRICKET$ is:

In how many ways can $15$ members of a council sit along a circular table,if the Secretary must sit on one side of the Chairman and the Deputy Secretary on the other side?