Show that 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 explanation:

Given that there are 101 people of different heights standing in a line

Consider the number 101

It is of the form

By application of Ramsey theory, any sequence consisting of distinct numbers contains a subsequence of length n+1 that is either decreasing or increasing.

Here we have 101 with n =10

Since 101 people are having different heights, we must have 10+1 heights i.e. heights of 11 people either increasing or decreasing.

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- 2 + 10x³ - 21x² + 19x - 4

step-by-step explanation:

each term in the second factor is multiplied by each term in the first factor

x²(- 2x² + 4x - 1) - 3x(- 2x² + 4x - 1) + 4(- 2x² + 4x - 1) ← distribute

= - 2 + 4x³ - x² + 6x³ - 12x² + 3x -8x² + 16x - 4

collect like terms

= - 2 + 10x³ - 21x² + 19x - 4 ← in standard form