Difference between revisions of "1986 AIME Problems/Problem 11"
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& = 1 + (1 - y) + (1 - y)^2 + (1 - y)^3 \cdots + (1 - y)^{17}\end{align*}.</cmath> | & = 1 + (1 - y) + (1 - y)^2 + (1 - y)^3 \cdots + (1 - y)^{17}\end{align*}.</cmath> | ||
| − | We want the coefficient of the <math>y^2</math> term of each power of each binomial, which by the binomial theorem is <math>{2\choose 2} + {3\choose 2} + \cdots + {17\choose 2}</math>. The [[Hockey Stick Identity]] gives us that this quantity is equal to <math>{18\choose 3} = 816</math>. | + | We want the coefficient of the <math>y^2</math> term of each power of each binomial, which by the binomial theorem is <math>{2\choose 2} + {3\choose 2} + \cdots + {17\choose 2}</math>. The [[Hockey Stick Identity]] gives us that this quantity is equal to <math>{18\choose 3} = \boxed{816}</math>. |
== See also == | == See also == | ||
Revision as of 19:20, 11 April 2012
Problem
The polynomial 
may be written in the form 
, where 
and the 
's are constants. Find the value of 
.
Solution
Solution 1
Using the geometric series formula, 
. Since 
, this becomes 
. We want , which is the coefficient of the 
term in 
(because the 
in the denominator reduces the degrees in the numerator by 
). By the binomial theorem that is 
.
Solution 2
Again, notice . So
We want the coefficient of the 
term of each power of each binomial, which by the binomial theorem is 
. The Hockey Stick Identity gives us that this quantity is equal to 
.
See also
| 1986 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
