Question

use pascals triangle to complete the expansion of $(v + w)^7$.
$v^7 + 7v^6w + 21v^5w^2 + 35v^4w^3 + \square v^3w^4 + 21v^2w^5 + 7vw^6 + w^7$

Explanation:

Step1: Recall Pascal's Triangle for \(n = 7\)

Pascal's Triangle rows correspond to binomial coefficients \(\binom{n}{k}\) for \((a + b)^n\). For \(n = 7\), the row of Pascal's Triangle (starting from \(k = 0\)) is: \(1, 7, 21, 35, 35, 21, 7, 1\).

Step2: Identify the coefficient for \(v^{3}w^{4}\)

In the expansion of \((v + w)^7\), the term \(v^{3}w^{4}\) has a coefficient equal to \(\binom{7}{4}\) (or \(\binom{7}{3}\), since \(\binom{n}{k}=\binom{n}{n - k}\)). From Pascal's Triangle, the 5th term (indexing from 0) in the row for \(n = 7\) is 35, which is the coefficient for \(v^{3}w^{4}\) (since the exponents of \(v\) start at 7 and decrease, so for \(v^{3}\), the term is at position \(k = 4\) in the binomial expansion \(\sum_{k = 0}^{7}\binom{7}{k}v^{7 - k}w^{k}\)).

Answer:

35