Question

use pascals triangle to complete the expansion of $(r + s)^5$. $r^5 + 5r^4s + 10r^3s^2 + \square r^2s^3 + 5rs^4 + s^5$

Explanation:

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

Pascal's Triangle rows correspond to binomial coefficients for \((a + b)^n\). The 5th row (starting from row 0) is \(1, 5, 10, 10, 5, 1\).

Step2: Identify the coefficient

In the expansion of \((r + s)^5\), the terms have coefficients from this row. The term \(r^2s^3\) corresponds to the 4th element (index 3, 0 - based) in the row, which is 10.

Answer:

10