Pascal's Triangle

History

100

What is the name of the triangle of numbers we are playing with today?

Pascal's Triangle.

200

In which country was Blaise Pascal born?

France

300

Is Pascal's Triangle a modern invention from the 21st century or was it studied hundreds of years ago?

Hundreds of years ago.

400

Did Pascal discover the triangle first, or was it already known in ancient Asia (like China and India)?

It was already known in ancient Asia.

500

What subject in mathematics uses Pascal's Triangle to calculate chances and possibilities?

Probability

Properties

100

What number is located at the top peak of Pascal's Triangle?

1

200

How do you get a number inside the triangle using the row above it?

By adding the two numbers directly above it

300

Are the numbers on the left side of the triangle the same as the numbers on the right side?

Yes

400

What are all the numbers on the outer borders (edges) of the triangle?

1s

500

If you add all numbers in Row 0 (1) you get 1. If you add Row 1 (1, 1) you get 2. What is the sum of Row 2 (1, 2, 1)?

4

Binomial Expansion

100

What is (x + y)¹?

(x + y)

200

What is (x + y)⁰?

1

300

In the expansion of (a + b)² = 1a² + 2ab + 1b², what are the three coefficients?

1, 2, 1

400

Which row in Pascal's Triangle gives the coefficients for (a + b)³?

Row 3 (1, 3, 3, 1)

500

If you expand (x + 1)², what is the middle number (coefficient)?

2

Patterns

100

What counting sequence appears in the second diagonal (1, 2, 3, 4, 5...)?

Natural numbers

200

If you look at Row 3 (1, 3, 3, 1), are all the numbers odd, or are some even?

All numbers are odd.

300

What simple pattern do you get if you multiply 11⁰, 11¹, and 11²?

The first rows of the triangle (1, 11, 121).

400

In Row 4 (1, 4, 6, 4, 1), which number is the even number?

4 and 6

500

What shape do you draw when you construct Pascal's Triangle?

A triangle

Applications & Curiosities

100

How many sides does a triangle have?

3

200

If you flip a coin, how many total possible outcomes are there?

2

300

If you flip two coins, how many total outcomes are possible (2²)?

4

400

Using Row 2 (1, 2, 1), in how many ways can you get exactly one Head when flipping two coins?

2 ways

500

How many total numbers are in the first 3 rows combined (Row 0, Row 1, and Row 2)?

6 numbers (1+2+3)