TED-Ed: The mathematical secrets of Pascal's triangle - Wajdi Mohamed Ratemi

Pascal's Triangle, known by various names across cultures, is a mathematical structure filled with patterns and secrets. It is generated by starting with 1 and adding pairs of numbers to form subsequent rows. Each row corresponds to the coefficients of a binomial expansion. The triangle reveals patterns like powers of 2 and geometric shapes such as triangular and tetrahedral numbers. It also has applications in probability and combinatorics, such as calculating probabilities of family compositions or forming basketball teams. The triangle continues to intrigue mathematicians, with recent expansions into new polynomial forms.

What is the primary method used to generate the rows of Pascal's Triangle?

• Dividing each number by its position

• Subtracting each number from the previous one

• Adding numbers in pairs with invisible zeros

• Multiplying each number by 2

Which mathematical concept is directly related to the rows of Pascal's Triangle?

• Geometric series

• Binomial expansion coefficients

• Prime numbers

• Fibonacci sequence

What pattern emerges when you sum the numbers in each row of Pascal's Triangle?

• Powers of 2

• Prime numbers

• Powers of 3

• Fibonacci numbers

What is the name of the fractal formed by shading odd numbers in Pascal's Triangle?

• Sierpinski's triangle

• Koch snowflake

• Mandelbrot set

• Cantor set

How can Pascal's Triangle be used to determine the number of ways to choose 5 players from a group of 12?

• By using the last element of row 12

• By using the sixth element of row 12

• By using the first element of row 12

• By using the sum of all elements in row 12