There are 5 parallel lines on the plane. On the same plane, there are $n$ other lines that are perpendicular to the 5 parallel lines. If the number of distinct rectangles formed by these lines is 360, what is the value of $n$?

A rabbit is sitting at the base of a staircase which has 10 steps. It proceeds to the top of the staircase by climbing either one step at a time or two steps at a time. The number of ways it can reach the top is

In a chess tournament, there are four groups, each containing an equal number of players. Each player plays

1. against every other player belonging to one's own group exactly once;
2. against each player belonging to one of the remaining three groups exactly twice;
3. against each player belonging to one of the remaining two groups exactly three times; and
4. against each player belonging to the remaining group exactly four times.

If there are more than 1000 matches being played in the tournament, the minimum possible number of players in each group is

In how many ways can the letters of the word MANAGEMENT be arranged such that no two vowels appear together?

The number of solutions of the equation $x_1 + x_2 + x_3 + x_4 = 50$, where $x_1, x_2, x_3, x_4$ are integers with $x_1 \geq 1, x_2 \geq 2, x_3 \geq 0, x_4 \geq 0$ is

Let $n$ be the number of ways in which 20 identical balloons can be distributed among 5 girls and 3 boys such that everyone gets at least one balloon and no girl gets fewer balloons than a boy does. Then

How many different numbers can be formed by using only the digits 1 and 3 which are smaller than 3000000?