$3-$

The city plan has a layout represented by a $5 \times 10$ rectangular grid. One-way traffic is in effect on the streets: travel is only allowed to the right and upwards. How many different routes are there from the bottom-left corner to the top-right corner?
Each route can be associated with a string of 15 letters "P" and "V" ("P" means moving one cell to the right, "V" - one cell up).

## Solution

Each route can be associated with a string of 15 letters "P" and "V" (see hint), with exactly 5 "P" letters and 10 "V" letters. Conversely, each such string of letters can uniquely determine a route. Therefore, the number of routes is equal to the number of such strings, which, in turn, is equal to $C_{15}^{5}$.

## Answer

3003 routes.

It is known that in a convex $n$-gon ($n>3$), no three diagonals pass through the same point.

Find the number of points (distinct from vertices) where pairs of diagonals intersect.

## Hint

Each pair of intersecting diagonals can be associated with a quadruple of vertices of the $n$-gon, which are the endpoints of these diagonals.

## Solution

The problem can be restated as follows: we need to find the number of pairs of intersecting diagonals of a convex $n$-gon. Each pair of intersecting diagonals can be associated with a quadruple of vertices of the $n$-gon, which are the endpoints of these diagonals. Conversely, for any quadruple of vertices, there is exactly one pair of intersecting diagonals whose endpoints are these four vertices. Thus, the number of pairs of intersecting diagonals is equal to the number of ways to choose four vertices from $n$.

## Answer

$C_{n}^{4}=\frac{n(n-1)(n-2)(n-3)}{24}$.