Pascal's Triangle and Binomial Theorem

Front

What is Pascal's Triangle?

Back

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

Front

What is the Binomial Theorem?

Back

The Binomial Theorem describes the algebraic expansion of powers of a binomial. It states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k for k = 0 to n.

Front

What does (n choose k) represent?

Back

(n choose k) or C(n, k) represents the number of ways to choose k elements from a set of n elements without regard to the order of selection.

Front

How do you find the third term in the expansion of (a + b)^n?

Back

The third term is given by the formula T(k+1) = (n choose k) * a^(n-k) * b^k, where k = 2 for the third term.

Front

What is the formula for the k-th term in the expansion of (a + b)^n?

Back

T(k+1) = (n choose k) * a^(n-k) * b^k.

Front

What is the expansion of (x + 1)^5?

Back

x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1.

Front

How do you determine the coefficients in the expansion of (a + b)^n?

Back

The coefficients are found in the n-th row of Pascal's Triangle.

Front

What is the fourth term in the expansion of (4x - 3)^5?

Back

-4320x^2.

Front

How do you find the 14th term in the expansion of (2x - y)^18?

Back

Use the formula T(k+1) = (n choose k) * (2x)^(n-k) * (-y)^k, where k = 13 for the 14th term.

Front

What is the significance of the row in Pascal's Triangle for (x + y)^3?

Back

The coefficients for the expansion of (x + y)^3 are found in the row 1 3 3 1 of Pascal's Triangle.