Question

use pascals triangle to complete the expansion of $(r + s)^4$.
$r^{4}+4r^{3}s + 6r^{2}s^{2}+\square rs^{\square}+s^{4}$

Explanation:

Step1: Recall Pascal's Triangle for power 4

Pascal's Triangle row for power \(n = 4\) is \(1,4,6,4,1\).

Step2: Analyze the binomial expansion formula \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\)

For \((r + s)^4\), when \(k = 3\), \(\binom{4}{3}r^{4-3}s^{3}\).
Since \(\binom{4}{3}=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=4\), and the exponent of \(s\) is \(3\) (because in the term \(rs^{\square}\), the sum of exponents of \(r\) and \(s\) must be \(4\), and if exponent of \(r\) is \(1\), then exponent of \(s\) is \(4-1 = 3\)).

Answer:

The coefficient is \(4\) and the exponent is \(3\).