Question

To prove if there are \(101\) people of different heights standing in a line, it is possible to find \(11\) people in the order they are standing in the line with heights that are either increasing or decreasing.

Short answer

Thus, if there are \(101\) people of different heights standing in a line, it is possible to find \(11\) people in the order they are standing in the line with heights that are either increasing or decreasing.

Step-by-step solution

Given data

\(101\) people of different heights standing in a line.

Concept used of number of ways to do a task

If a task is complete in\(n1\)ways or\(n2\) ways, then the number of ways to do the task is\(n1 + n2\).

Prove the statement

There are \(101\) people, standing in a line and all of them have different heights.

Since, \(101 = {10^2} + 1\), by Ramsey theory, every sequence consisting of \({n^2} + 1\) distinct real numbers contains a subsequence of length \(n + 1\) that is either strictly increasing or decreasing.

Put \(n = 10\), it will be proved that it is possible to find \(11\) people among those \(101\) whose heights are either decreasing or increasing.

Thus, if there are \(101\) people of different heights standing in a line, it is possible to find \(11\) people in the order they are standing in the line with heights that are either increasing or decreasing.