Number of shortest paths from $HOSTEL$ to $ALLEN$ is equal to (as shown in the given figure):

How many $3$-digit numbers can be formed using $0, 2, 3, 5, 7$ when repetition is allowed?

$A$ box contains two white balls,three black balls,and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?

In a certain test,there are $n$ questions. In the test,$2^{n-i}$ students gave wrong answers to at least $i$ questions,where $i = 1, 2, \dots, n$. If the total number of wrong answers given is $2047$,then $n$ is equal to

In how many ways can a team of $11$ players be formed out of $25$ players,if $6$ out of them are always to be included and $5$ are always to be excluded?

Let $n$ be the number of different $5$-digit numbers,divisible by $4$,formed with the digits $1, 2, 3, 4, 5,$ and $6$,such that no digit is repeated in the numbers. What is the value of $n$?