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Use Pascal's Triangle to complete the expansion of $\newline$ $(p + q)^6$ . $p^6 +$ ____ $p^5q + 15p^4q^2 + 20p^3q^3 + 15p^2q^4 + 6pq^5 + q^6$

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Pascal's triangle and the Binomial Theorem

Full solution

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

\newline

(p + q)^6

.

p^6 +

____

p^5q + 15p^4q^2 + 20p^3q^3 + 15p^2q^4 + 6pq^5 + q^6

Explanation: Pascal's Triangle gives us the coefficients for the expansion of $(p + q)^n$ . For $n=6$ , the row is $1, 6, 15, 20, 15, 6, 1$ .
Identifying Missing Term: We already have the coefficients for $p^6$ , $p^4q^2$ , $p^3q^3$ , $p^2q^4$ , $pq^5$ , and $q^6$ . We need the coefficient for $p^5q$ .
Finding Coefficient: The coefficient for $p^5q$ is the second number in the $6$ th row of Pascal's Triangle, which is $6$ .
Final Answer: So the missing term is $6p^5q$ .

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