Pascal's Triangle and Binomial Expansion

Front

What is Pascal's Triangle?

Back

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two directly above it. It is used to find coefficients in binomial expansions.

Front

What is the formula for binomial expansion?

Back

The binomial expansion of (a + b)^n is given by: \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k, where \binom{n}{k} is a binomial coefficient.

Front

What is the binomial coefficient \binom{n}{k}?

Back

The binomial coefficient \binom{n}{k} represents the number of ways to choose k elements from a set of n elements, calculated as \frac{n!}{k!(n-k)!}.

Front

How do you find the coefficients for (x + y)^4 using Pascal's Triangle?

Back

The coefficients for (x + y)^4 are found in the 4th row of Pascal's Triangle: 1, 4, 6, 4, 1.

Front

Expand (x + 5)^2.

Back

(x + 5)^2 = x^2 + 10x + 25.

Front

What is the 3rd row of Pascal's Triangle?

Back

The 3rd row of Pascal's Triangle is: 1, 3, 3, 1.

Front

Expand (a + b)^3.

Back

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.

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

Expand (x + 5)^5.

Back

(x + 5)^5 = x^5 + 25x^4 + 250x^3 + 1250x^2 + 3125x + 3125.

Front

What pattern do you observe in the rows of Pascal's Triangle?

Back

Each row corresponds to the coefficients of the binomial expansion of (a + b)^n, where n is the row number starting from 0.