using pascals teiangle what is the third term in the expansion of (a+b)^5 when the expanded polynomial is written is standard form

Answer 1

To find the third term in the expansion of \((a + b)^5\) using Pascal's Triangle, we can start by identifying the coefficients from the triangle.

The coefficients for the expansion of \((a + b)^n\) are given by the \(n\)-th row of Pascal's Triangle. For \(n = 5\), the coefficients are:

\[ 1, 5, 10, 10, 5, 1 \]

The general term in the binomial expansion is given by:

\[ T_k = \binom{n}{k} a^{n-k} b^k \]

where \(k\) starts from 0. The third term corresponds to \(k = 2\) (since we start counting from \(k = 0\)).

Using \(n = 5\) and \(k = 2\):

1. Calculate the binomial coefficient: \[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

2. Write the third term: \[ T_2 = \binom{5}{2} a^{5-2} b^2 = 10 a^{3} b^{2} \]

So, the third term in the expansion of \((a + b)^5\) is:

\[ 10a^3b^2 \]