Pascal's Triangle & Binomial Expansion

Front

What is Pascal's Triangle?

Back

A triangular array of the binomial coefficients, where each number is the sum of the two directly above it.

Front

How do you create Pascal's Triangle?

Back

Start with 1 at the top, place 1's on the outside edges, and each interior number is the sum of the two numbers above it.

Front

What are the coefficients of the 5th row in Pascal's Triangle?

Back

1;5;10;10;5;1

Front

Which row of Pascal's Triangle corresponds to the expansion of (x+y)^3?

Back

1;3;3;1

Front

What is the formula for the nth row of Pascal's Triangle?

Back

The nth row corresponds to the coefficients of (a+b)^n, where n is the row number.

Front

What is the relationship between Pascal's Triangle and binomial expansion?

Back

The coefficients in the binomial expansion of (a+b)^n are given by the nth row of Pascal's Triangle.

Front

What is the sum of the numbers in the nth row of Pascal's Triangle?

Back

2^n, where n is the row number.

Front

How can you find the number of combinations using Pascal's Triangle?

Back

The number in the nth row and kth position represents the number of combinations of n items taken k at a time.

Front

What is the significance of the diagonal rows in Pascal's Triangle?

Back

The diagonals represent the triangular numbers, which count objects arranged in an equilateral triangle.

Front

What is the pattern of even and odd numbers in Pascal's Triangle?

Back

The even and odd numbers create a fractal pattern known as the Sierpinski triangle.