HISTORY
Who is Pascal’s Triangle named after?
It’s named after Blaise Pascal, a French mathematician, physicist, and philosopher who studied and published work on the triangle in 1654.
What are the first and last numbers in every row?
They are always 1.
What type of sequence do the first numbers in each row make?
All 1s.
How can Pascal’s Triangle help with flipping coins?
It shows the number of ways heads and tails can appear.
What happens if you color all even numbers in the triangle?
It forms a cool fractal (Sierpinski triangle)
Why is Pascal’s Triangle important?
Binomial coefficients
Probability theory
Combinatorics (counting combinations)
Patterns in numbers
Pascal used it to develop early ideas of probability that helped form modern statistics.
What is the basic structure of Pascal’s Triangle?
Each number is the sum of the two numbers directly above it.
What pattern do you get when you add all the numbers in a row?
Powers of 2 (1, 2, 4, 8, 16…)
In which branch of math is Pascal’s Triangle most used?
Probability and combinatorics.
What famous sequence appears when you add up the diagonals?
The Fibonacci sequence.
What was Pascal’s main work involving the triangle called?
His main work was called “Traité du triangle arithmétique” (Treatise on the Arithmetical Triangle), published in 1654.
What is the sum of the numbers in the nth row?
The sum of all numbers in the nth row is 2ⁿ.
Example: Row 3 → 1 + 3 + 3 + 1 = 8 = 2³.
What diagonal in the triangle represents the counting numbers (1, 2, 3, 4…)?
The third diagonal.
How can Pascal’s Triangle help you find powers of binomials like (a + b)⁵?
Each row gives the coefficients for expansion.
Where else (besides math) has Pascal’s Triangle appeared — music, art, or architecture?
In art and design patterns (symmetry, mosaics, etc.).
What modern uses come from Pascal’s studies?
Binomial theorem in algebra
Binomial distribution in probability
Computer algorithms
Fractals and patterns in nature
How can you identify a prime number using Pascal’s Triangle?
If n is prime, then all the numbers in the nth row (except the 1s) are divisible by that prime.
What pattern do you see if you color even and odd numbers differently?
A repeating triangular fractal pattern.
How is Pascal’s Triangle connected to probability trees?
The triangle gives the number of possible outcomes at each level.
What’s a fun fact about Pascal’s Triangle and hockey sticks?
The “hockey stick pattern” shows that adding a diagonal line of numbers equals the number below them.