Basics
Who is Blaise Pascal?
This French mathematician gives his name to the triangle.
What is 1?
Each row starts and ends with this number.
Each number in Pascal’s Triangle represents this mathematical concept.
A combination (n choose k)
Pascal’s Triangle gives the coefficients for expanding this type of expression.
A binomial (a + b)ⁿ
Pascal’s Triangle was studied long before Pascal by this group of mathematicians.
Chinese mathematicians
What is 1?
The number at the very top of Pascal’s Triangle.
The second row is made up of these two numbers.
1 and 1
The middle numbers in each row correspond to the largest values of this operation.
Combinations
The coefficients for (a + b)³ are 1, 3, 3, 1. Write the expansion.
a³ + 3a²b + 3ab² + b³
The triangle can be used to find powers of this number pattern (like 11¹ = 11, 11² = 121).
11
What are the two numbers directly above it?
Each number in the triangle is found by adding these two numbers above it.
The third row (1, 2, 1) represents powers of what number?
2
The fifth row corresponds to these combinations: 5C0, 5C1, 5C2, 5C3, 5C4, 5C5.
1, 5, 10, 10, 5, 1
The second term in any binomial expansion can be found by multiplying the first coefficient by this.
n
This diagonal of the triangle lists the counting numbers: 1, 2, 3, 4, 5...
The second diagonals
The triangle starts with row 0 or 1?
row 0
The diagonals of Pascal’s Triangle show these famous number patterns.
Natural numbers, triangular numbers, etc.
The formula for a combination.
n! / (k!(n−k)!)
In (x + y)⁵, the fourth term has this coefficient.
10
Pascal’s Triangle connects to this famous mathematical sequence: 1, 1, 2, 3, 5, 8...
Fibonacci sequence
What is addition?
The triangle can be built using this simple mathematical operation.
The sum of the numbers in each row equals this mathematical expression.
2ⁿ
The row number n represents the coefficients of a binomial raised to this power.
n
The sum of all coefficients in a binomial expansion equals this value.
2n
The triangle can be used in probability, algebra, and this type of mathematical study.
Combinatorics or geometry