x_(A) Use Pascal's Triangle to complete the expansion of (r+s)^4 r^4+square r^3s+6r^2s^2+square rs^3+s^4

To expand $(r+s)^{4}$ using Pascal's Triangle, we can use the binomial theorem. The binomial theorem states that $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$, where $\binom{n}{k}$ is the binomial coefficient given by $\frac{n!}{k!(n-k)!}$.<br /><br />For $(r+s)^{4}$, we look at the 4th row of Pascal's Triangle (remembering that we start counting rows from row 0), which is 1, 4, 6, 4, 1. These are the coefficients we will use.<br /><br />The expansion is:<br />$(r+s)^{4} = 1r^{4} + 4r^{3}s + 6r^{2}s^{2} + 4rs^{3} + 1s^{4}$<br /><br />So, the completed expansion is:<br />$r^{4} + 4r^{3}s + 6r^{2}s^{2} + 4rs^{3} + s^{4}$