$A$ four-digit number is to be formed using the digits $1, 2, 3, 4, 5, 6, 7$ (no digit is repeated). What is the probability that the number formed is $> 4000$?

If the letters of the word "$UNIVERSITY$" are arranged at random,what is the probability that both '$I$'s do not come together (in $/5$)?

An old man forgets the last two digits of a telephone number. He dials these two digits randomly. What is the probability that he dials the correct number?

Out of $3$ men,$2$ women,and $4$ children,if $4$ people are selected at random,what is the probability that exactly $2$ children are selected?

Twenty tickets are marked with the numbers $1, 2, \dots, 20$. If three tickets are drawn at random,what is the probability that the tickets marked $7$ and $11$ are among them?

Five persons enter a lift cabin in the cellar of a $7$-floor building (excluding the cellar). If each of them independently and with equal probability can leave the cabin at any floor out of the $7$ floors (beginning with the first), then the probability that all $5$ persons leave the cabin at different floors is: