Difference between revisions of "1998 AIME Problems/Problem 7"
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== Problem == | == Problem == | ||
| + | Let <math>n</math> be the number of ordered quadruples <math>\displaystyle(x_1,x_2,x_3,x_4)</math> of positive odd [[integer]]s that satisfy <math>\sum_{i = 1}^4 x_i = 98.</math> Find <math>\frac n{100}.</math> | ||
== Solution == | == Solution == | ||
| + | Define <math>\displaystyle x_i = 2y_i - 1</math>. Then <math>2\left(\sum_{i = 1}^4 y_i\right) - 4 = 98</math>, so <math>\sum_{i = 1}^4 y_i = 51</math>. | ||
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| + | So we want to find four integers that sum up to 51; we can imagine this as trying to split up 51 on the number line into 4 ranges. This is equivalent to trying to place 3 markers on the numbers 1 through 50; thus the answer is <math>n = {50\choose3} = \frac{50 * 49 * 48}{3 * 2} = 19600</math>, and <math>\frac n{100} = 196</math>. | ||
== See also == | == See also == | ||
| − | + | {{AIME box|year=1998|num-b=6|num-a=8}} | |
| + | |||
| + | [[Category:Intermediate Combinatorics Problems]] | ||
Revision as of 21:38, 7 September 2007
Problem
Let 
be the number of ordered quadruples 
of positive odd integers that satisfy 
Find 
Solution
Define 
. Then 
, so 
.
So we want to find four integers that sum up to 51; we can imagine this as trying to split up 51 on the number line into 4 ranges. This is equivalent to trying to place 3 markers on the numbers 1 through 50; thus the answer is 
, and 
.
See also
| 1998 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
