The value of ${}^7P_3 - 3({}^6P_2)$ is equal to

We are to form different words with the letters of the word $INTEGER$. Let $m_1$ be the number of words in which $I$ and $N$ are never together and $m_2$ be the number of words which begin with $I$ and end with $R$,then $m_1/m_2$ is equal to

In a class of $20$ students,in how many ways can the following prizes be awarded?
If the first and second prizes are for Mathematics,the first and second prizes for Physics,the first prize for Chemistry,and the first prize for English are to be awarded.

The number of all $5$-digit numbers with distinct digits is:

The number of four-digit numbers that can be formed using the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ which are divisible by $3$,when repetition of digits is allowed any number of times,is

$10$ men and $6$ women are to be seated in a row so that no two women sit together. The number of ways they can be seated is: