If $(a+b+c+d+e+f+g+h+i)^{2}$ is expanded and simplified, how many different terms are in the final answer?
(A) 36
(B) 9
(C) 45
(D) 81
(E) 72
If $(a+b+c+d+e+f+g+h+i)^{2}$ is expanded and simplified, how many different terms are in the final answer?
(A) 36
(B) 9
(C) 45
(D) 81
(E) 72

## Solution

$$
\begin{gathered}
\text { Bracket 1 } \\
(a+b+c+d+e+f+g+h+i)(a+b+c+d+e+f+g+h+i)
\end{gathered}
$$

If we wish to determine how many different terms can be produced we begin by multiplying the ' $a$ ' in bracket 1 by each term in bracket 2 . This calculation gives 9 different terms. We continue this process by now multiplying the ' $b$ ' in bracket 1 by the elements from $b$ to $i$ in bracket 2 to give 8 different terms. We continue this process until we finally multiply the ' $i$ ' in the first bracket by the ' $i$ ' in the second bracket. Altogether we have, $9+8+7+6+5+4+3+2+1=45$ different terms.

ANSWER: (C)