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Use Pascal's Triangle to complete the expansion of $(x + y)^5$ . $\newline$ $x^5 + 5x^4y + 10x^3y^2 + \underline{\quad}x^2y^3 + 5xy^4 + y^5$

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Pascal's triangle and the Binomial Theorem

Full solution

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

(x + y)^5

.

\newline

x^5 + 5x^4y + 10x^3y^2 + \underline{\quad}x^2y^3 + 5xy^4 + y^5

Identify Pascal's Triangle: Pascal's Triangle for the $5^{\text{th}}$ row is $1, 5, 10, 10, 5, 1$ . We already have the coefficients for $x^5, x^4y,$ and $x^3y^2$ . We need the coefficient for $x^2y^3$ .
Find Coefficient: The coefficient for $x^2y^3$ is the fourth number in the $5$ th row of Pascal's Triangle, which is $10$ .
Write Complete Expansion: Now we write the complete expansion: $x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5$ .

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