Pascals triangle and binomial expansion

Front

What is Pascal's Triangle?

Back

Pascal's Triangle is a triangular array of the binomial coefficients, where each number is the sum of the two directly above it.

Front

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

Back

1; 5; 10; 10; 5; 1

Front

How do you expand (x + 2)^3 using the binomial theorem?

Back

x^3 + 6x^2 + 12x + 8

Front

What is a binomial?

Back

A binomial is a polynomial that consists of the sum or difference of two terms.

Front

What is the formula for the binomial expansion of (a + b)^n?

Back

The formula is (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k = 0 to n.

Front

What does 'n choose k' represent in binomial coefficients?

Back

'n choose k' represents the number of ways to choose k elements from a set of n elements, calculated as n! / (k!(n-k)!).

Front

What is the relationship between the rows of Pascal's Triangle and binomial coefficients?

Back

Each row of Pascal's Triangle corresponds to the coefficients of the expanded form of (a + b)^n.

Front

How can you find the coefficients of the 6th row in Pascal's Triangle?

Back

The coefficients of the 6th row are found by calculating '6 choose k' for k = 0 to 6, resulting in 1; 6; 15; 20; 15; 6; 1.

Front

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

Back

The sums of the diagonals in Pascal's Triangle correspond to the Fibonacci sequence.

Front

How do you calculate the 4th power of a binomial (x + y)^4?

Back

Using the binomial theorem: (x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4.