Intervals

Probability Level 5

Consider a row of tiles, numbered from 1 to 10 000.

By an interval we mean one or more consecutive tiles. For instance, there are precisely 6001 intervals of length 4000.

The $1729^\text{th}$ tile is marked with a red cross.

How many intervals are there that contain the tile with the cross?

1 solution

Arjen Vreugdenhil
Apr 27, 2016

General solution for $n$ tiles, with the cross on the $a$ th tile:

The first tile of the interval is any tile between 1 and $a$ ; this gives $a$ choices.

The last tile is any tile between $a$ and $n$ ; this gives $n+1-a$ choices.

Since the tiles can be chosen independent of each other, there are $N = a(n+1-a)$ intervals containing the red cross.

Here, $N = 1729\cdot (10\:001-1729) = \boxed{14\:302\:288}$ .