View source

==The Triangle==

Here are lines zero through eight of Pascal's triangle:

 Row
 0:                           1                                              
 1:                        1     1                                           
 2:                     1     2    1                                         
 3:                  1     3     3    1                                      
 4:               1     4     6     4    1                                   
 5:            1     5     10    10    5    1                                
 6:         1     6     15    20   15    6     1                             
 7:      1     7     21    35    35    21    7    1                          
 8:   1     8     28    56    70    56    28    8    1                       
<!-- 9:     1     9     36    84    126   126   84    36    9    1                    
 10: 1     10    45    120   210   252   210   120   45    10   1                 
 11:   1     11    55   165   330    462   462   330   165   55    11  1               
 12:1     12    66    220   495   792   924   792   495   220   66    12  1-->        
 
== Introductory ==
=== Problem 1 ===

Given <math>\displaystyle (x + y)^{10}</math>, find:

# The coefficient of the <math>x^{4}y^{6}</math> term.
# The sum of the coefficients.

<br>'''Solution'''

1. You need to find the 6th number (remember the first number in each row is considered the 0th number) of the 10th row in Pascal's triangle.

The 10th row is:

 1  10  45  120  210  252  210  120  45  10  1

Thus the coefficient is the 6th number in the row or <math>\displaystyle{210}</math>.

This can also be found using the binomial theorem:

::<math>(x+y)^{k}=\sum^{k}_{n=0}\displaystyle{k\choose n}x^{n}y^{k-n}</math>

Through the summation, the binomial theorem will provide you with the coefficient if each term of the result.  In our particular case, we are only looking for the coefficient of the <math>\displaystyle{x^{4}y^{6}}</math> term.

Since you are looking for <math>\displaystyle{x^{4}y^{6}}</math> term in <math>\displaystyle{(x+y)^{10}}</math>, then <math>\displaystyle{n=4}</math> and <math>\displaystyle{k=10}</math>.

So the coefficient of the <math>\displaystyle{x^{4}y^{6}}</math> term is <math>\displaystyle{10\choose 4}=210</math>.

2.  Since all the coefficients are found in the 10th row, we simply need to add the numbers in the 10th row together.  This can be done by hand since there are relatively few numbers, but we could also use the following formula to sum up the numbers:

::<math>\sum_{n=0}^{k}{k \choose n}=2^k</math>

This summation formula simply adds up all the coefficients since <math>\displaystyle{k\choose n}</math> gives us each of the coefficients.  So, the sum is <math>2^{10}=1024</math>.

For your information, the final polynomial which results from <math>(x+y)^{10}</math> is <math>\displaystyle x^{10} + 10x^9y + 45x^8y^2 + 120x^7y^3 + 210x^6y^4 + 252x^5y^5</math><math> + 210x^4y^6 + 120x^3+y^7 + 45x^2+y^8+10x^1y^9+y^{10}</math>

== Intermediate ==
=== Problem 1 ===
<div style="text-align:right;">[[1992 AIME Problems/Problem 4|1992 AIME, # 4]]</div>
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 <math>3: 4: 5</math>?

[[1992 AIME Problems/Problem 4|Solution]]

=== Problem 2 ===
<div style="text-align:right;">Source: [[2007 AIME II Problems/Problem 13|2007 AIME II, # 13]]</div>
A [[triangle|triangular]] [[array]] of [[square]]s has one square in the first row, two in the second, and in general, <math>k</math> squares in the <math>k</math>th row for <math>1 \leq k \leq 11.</math> With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a <math>0</math> or a <math>1</math> is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of <math>0</math>'s and <math>1</math>'s in the bottom row is the number in the top square a [[multiple]] of <math>3</math>?

[[Image:2007 AIME II-13.png]]

[[2007 AIME II Problems/Problem 13|Solution]]

== Olympiad ==
{{stub}}

[[Category:Introductory Combinatorics Problems]] [[Category:Intermediate Combinatorics Problems]]