How many $3$-digit numbers can be formed using $0, 1, 3, 5, 7$ when repetition is allowed?

The number of arrangements of the letters of the word $BANANA$ in which two $N$s do not appear adjacently is

Each section in the first year of a plus $2$ course has exactly $40$ students. If there are $5$ sections,in how many ways can a set of $4$ student representatives be selected,$1$ from each section?

In how many different ways can the letters of the word '$OPTICAL$' be arranged so that the vowels always come together?

Let $n > 2$ be an integer. Suppose that there are $n$ Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track. Further,each pair of nearest stations is connected by a blue line,whereas all remaining pairs of stations are connected by a red line. If the number of red lines is $99$ times the number of blue lines,then the value of $n$ is

The sum $\sum_{r=1}^{10} (r^2 + 1) \times (r!)$ is equal to