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Use Pascal's Triangle to complete the expansion of $(u + v)^4$ . $\newline$ $u^4 + 4u^3v + \underline{\hspace{2em}}u^2v^2 + 4uv^3 + v^4$

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Pascal's triangle and the Binomial Theorem

Full solution

Q. Use Pascal's Triangle to complete the expansion of

(u + v)^4

.

\newline

u^4 + 4u^3v + \underline{\hspace{2em}}u^2v^2 + 4uv^3 + v^4

Pascal's Triangle Coefficients: Pascal's Triangle for the $4$ th row is $1, 4, 6, 4, 1$ . These numbers are the coefficients for the expansion.
Identifying Missing Term: The missing term corresponds to the $u^2v^2$ term in the expansion, which is the third term.
Calculating Missing Term: Using the $3^{\text{rd}}$ coefficient from Pascal's Triangle, which is $6$ , the missing term is $6u^2v^2$ .

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