1998 AIME Problems/Problem 7: Difference between revisions
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== Problem == | == Problem == | ||
Let <math>n</math> be the number of ordered quadruples <math> | Let <math>n</math> be the number of ordered quadruples <math>(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 == | ||
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[[Category:Intermediate Combinatorics Problems]] | [[Category:Intermediate Combinatorics Problems]] | ||
{{MAA Notice}} | |||
Revision as of 18:38, 4 July 2013
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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing
