1958 AHSME Problems/Problem 49
Problem
In the expansion of $(a \plus{} b)^n$ (Error compiling LaTeX. Unknown error_msg) there are $n \plus{} 1$ (Error compiling LaTeX. Unknown error_msg) dissimilar terms. The number of dissimilar terms in the expansion of $(a \plus{} b \plus{} c)^{10}$ (Error compiling LaTeX. Unknown error_msg) is:
Solution
Expand the binomial 
with the binomial theorem. We have:
![\[\sum\limits_{k=0}^{10} \binom{10}{k} (a+b)^k c^{10-k}\]](http://latex.artofproblemsolving.com/e/7/2/e725bca22403df696716c07255dd306e9dcd4333.png)
So for each iteration of the summation operator, we add k+1 dissimilar terms. Therefore our answer is:
![\[\sum\limits_{k=0}^{10} k+1 = \frac{11(1+11)}{2} = 66 \to \boxed{\textbf{D}}\]](http://latex.artofproblemsolving.com/b/d/c/bdc92d46f69d4d77597bf6b65ef54c6d80639b20.png)
See also
| 1958 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 48 |
Followed by Problem 50 | |
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| All AHSME Problems and Solutions | ||
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