Question

1. what is the coefficient of $x^3y^2$ in the expansion of $(x + y)^5$ using pascal’s triangle?

a. 20
b. 10
c. 5
d. 15

Explanation:

Step1: Recall Pascal's Triangle for \(n = 5\)

Pascal's Triangle rows start with \(n = 0\). For \(n = 5\), the row is \(1, 5, 10, 10, 5, 1\).

Step2: Identify the term's position

In the expansion of \((x + y)^5=\sum_{k = 0}^{5}\binom{5}{k}x^{5 - k}y^{k}\), for \(x^{3}y^{2}\), we have \(5 - k=3\) (so \(k = 2\)). The coefficient is the 3rd element (index starting at 0) in the \(n = 5\) row, which is \(10\).

Answer:

b. 10