Binomial Theorem and Pascal's Triangle

Front

What is the Binomial Theorem?

Back

The Binomial Theorem provides a formula for the expansion of powers of a binomial, expressed as (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n.

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. It is used to find coefficients in binomial expansions.

Front

How do you find the nth term in a binomial expansion?

Back

The nth term in the expansion of (a + b)^n is given by T(k+1) = (n choose k) * a^(n-k) * b^k, where k is the term number starting from 0.

Front

What is the formula for combinations (n choose k)?

Back

The formula for combinations is (n choose k) = n! / (k!(n-k)!), where n! is the factorial of n.

Front

What is the 2nd term in the expansion of (2b + 1)^3?

Back

12b^2.

Front

What is the 6th term in the expansion of (3x + 2)^8?

Back

48384x^3.

Front

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

Back

The 3rd row: 1, 3, 3, 1.

Front

What is the third term in the expansion of (2x - 5)^5?

Back

2000x^3.

Front

How many rows are there in Pascal's Triangle?

Back

Infinite.

Front

What is the significance of the coefficients in Pascal's Triangle?

Back

The coefficients represent the number of ways to choose k elements from a set of n elements, which corresponds to the binomial expansion.