$4$ buses run between Bhopal and Gwalior. If a man goes from Gwalior to Bhopal by a bus and comes back to Gwalior by a different bus,then the total number of possible ways is:

Find the value of $n$ such that $^{n}P_{5} = 42 \, ^{n}P_{3}$,where $n > 4$.

How many ways are there to arrange the letters of the word $EDUCATION$ so that all the following three conditions hold?
- The vowels occur in the same order $(E, U, A, I, O)$.
- The consonants occur in the same order $(D, C, T, N)$.
- No two consonants are next to each other.

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 $.....$

The number of natural numbers less than $1000$,in which no two digits are repeated,is:

$A$ number lock consists of three rings. If the $1^{st}$ ring is marked with digits $0$ to $9$,the $2^{nd}$ ring is marked with prime numbers greater than $2$ but less than $30$,and the $3^{rd}$ ring is marked with all vowels,find the total number of unsuccessful attempts.