2012 AMC 12A Problems/Problem 20
Contents
Problem
Consider the polynomial
The coefficient of is equal to . What is ?
Solution 1
Every term in the expansion of the product is formed by taking one term from each factor and multiplying them all together. Therefore, we pick a power of or a power of from each factor.
Every number, including , has a unique representation by the sum of powers of two, and that representation can be found by converting a number to its binary form. , meaning .
Thus, the term was made by multiplying from the factor, from the factor, and so on. The only numbers not used are , , and .
Thus, from the factors, , , and were chosen as opposed to , and .
Thus, the coefficient of the term is . So the answer is .
Solution 2
The degree of is . We want to find the coefficient of , so we need to omit the powers of that add up to . We find that . From here, we know that the answer is . Therefore, the answer is
See Also
2012 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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