Question

Is the binomial theorem and pascal's formula the same?

Answer 1

The binomial theorem and Pascal's formula are related but not the same. The binomial theorem allows us to expand a binomial expression, while Pascal's formula relates the coefficients of the terms in the expansion to Pascal's triangle.

Step-by-step explanation:

The binomial theorem and Pascal's formula are related concepts but they are not the same. The binomial theorem is a formula that allows us to expand a binomial expression raised to a power. It gives us the coefficients of the expanded terms. Pascal's formula, on the other hand, relates the coefficients of the terms in the expansion of a binomial expression to the numbers in Pascal's triangle.

For example, if we have the expression (a + b)^3, the binomial theorem tells us that it expands to a^3 + 3a^2b + 3ab^2 + b^3. Pascal's formula tells us that the coefficients of the terms in this expansion are 1, 3, 3, and 1, which can be found in the third row of Pascal's triangle.

Answer 2

They are closely related. The Binomial Theorem expands (a+b)ⁿ as the sum of a series of terms involving progressive powers of a and b with a coefficient based entirely on the exponent n. Coefficients of consecutive terms in the series are the same as the nth rows of Pascal’s Triangle. This means that the formula used for calculating the coefficients in the Binomial expansion is the same one used to calculate the numbers in the corresponding row of Pascal’s Triangle. The nomenclature ⁿcᵣ for the number of ways of combining r objects out of n objects invokes this formula.