Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1992 AIME Problems/Problem 4"
m
Line 5: Line 5:
 
== Solution ==
 
== Solution ==
  
−
In Pascal's Triangle, we know that the binomial coefficients of the nth row are <math>\displaystyle\binom{n}{0}</math>, <math>\displaystyle \binom{n}{1}</math>,...,<math>\displaystyle \binom{n}{n}</math>. Let our row be the nth row such that the three consecutive entries are <math>\displaystyle \binom{n}{r}</math>, <math>\displaystyle \binom{n}{r+1}</math>, and <math>\displaystyle \binom{n}{r+2}</math>.  
+
In Pascal's Triangle, we know that the binomial coefficients of the <math>n</math>th row are <math>n\choose 0</math>, <math>n\choose1</math>, ..., <math>n\choose n</math>. Let our row be the <math>n</math>th row such that the three consecutive entries are <math>n \choose r</math>, <math>n \choose r+1</math> and <math>n \choose r+2</math>.  
  
−
After expanding and dividing one entry by another (to clean up the factorials), we see that <math>\displaystyle \frac 34=\frac{r+1}{n-r}</math> and <math>\displaystyle \frac45=\frac{r+2}{n-r-1}</math>. Solving, <math>\displaystyle n = \boxed{062}</math>
+
After expanding and dividing one entry by another (to clean up the factorials), we see that <math>\displaystyle \frac 34=\frac{r+1}{n-r}</math> and <math>\displaystyle \frac45=\frac{r+2}{n-r-1}</math>. Solving, <math>\displaystyle n = 062</math>
−
'''
 
−
SOMEONE PLEASE SAVE THE LATEX!'''
 
  
 
== See also ==
 
== See also ==
Line 17: Line 15:
  
 
* [[1992 AIME Problems]]
 
* [[1992 AIME Problems]]
 +
 +
[[Category:Intermediate Combinatorics Problems]]
 +
[[Category:Intermediate Number Theory Problems]]

Revision as of 20:20, 6 November 2006

Problem

In Pascal's Triangle, each entry is the sum of the two entries above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$?

Solution

In Pascal's Triangle, we know that the binomial coefficients of the $n$th row are $n\choose 0$, $n\choose1$, ..., $n\choose n$. Let our row be the $n$th row such that the three consecutive entries are $n \choose r$, $n \choose r+1$ and $n \choose r+2$.

After expanding and dividing one entry by another (to clean up the factorials), we see that $\displaystyle \frac 34=\frac{r+1}{n-r}$ and $\displaystyle \frac45=\frac{r+2}{n-r-1}$. Solving, $\displaystyle n = 062$

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1992_AIME_Problems/Problem_4&oldid=11303"