The number of ways in which the following prizes can be given to a class of $20$ boys,first and second Mathematics,first and second Physics,first Chemistry,and first English is:

If $\frac{^{56}P_{r+6}}{^{54}P_{r+3}} = 30800$,then $r = \dots$

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

The letters of the word '$MANKIND$' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word '$MANKIND$' is $.....$

How many numbers less than $1000$ can be formed using the digits $1, 2, 3, 4, 5, 6$ if repetition is not allowed?

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.