The Binomial Theorem

Front

What is the Binomial Theorem?

Back

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer.

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 is row 7 of Pascal's Triangle?

Back

1, 7, 21, 35, 35, 21, 7, 1

Front

How do you calculate a binomial coefficient?

Back

The binomial coefficient C(n, k) is calculated using the formula C(n, k) = n! / (k!(n-k)!), where n is the total number of items, and k is the number of items to choose.

Front

What is the general term in the expansion of (a + b)^n?

Back

The general term is given by T(k+1) = C(n, k) * a^(n-k) * b^k, where k = 0, 1, 2, ..., n.

Front

What is the significance of the coefficients in the expansion of (a + b)^n?

Back

The coefficients correspond to the values in row n of Pascal's Triangle.

Front

Expand (x + 2)^3 using the Binomial Theorem.

Back

(x + 2)^3 = x^3 + 3(2)x^2 + 3(2^2)x + 2^3 = x^3 + 6x^2 + 12x + 8.

Front

What is the value of C(5, 2)?

Back

C(5, 2) = 5! / (2!(5-2)!) = 10.

Front

What is the relationship between the Binomial Theorem and combinatorics?

Back

The Binomial Theorem is used in combinatorics to find the number of ways to choose k elements from a set of n elements.

Front

What is the expansion of (a - b)^4?

Back

(a - b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4.