Question

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

Explanation:

Step1: Recall binomial expansion formula

The binomial expansion of \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\). For \((q + r)^5\), when \(k = 0\), \(a=q\), \(b = r\), \(n = 5\).

Step2: Calculate the first term

Using the binomial coefficient \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\), when \(n = 5\) and \(k=0\), \(\binom{5}{0}=\frac{5!}{0!(5 - 0)!}=\frac{5!}{5!×1}=1\), and \(q^{5-0}r^{0}=q^{5}\). So the first term is \(1\times q^{5}\times r^{0}=q^{5}\).

Answer:

\(q^{5}\)