+0  
 
+1
75
3
avatar+2610 

1. Find the coefficient of \(x^3y^3z^2\) in the expansion of \((x+y+z)^8\).

 

2. For each integer \(n\), let \(f(n)\) be the sum of the elements of the \(n\) th row (i.e. the row with \(n+1\) elements) of Pascal's triangle minus the sum of all the elements from previous rows. For example, \(f(2) = \underbrace{(1 + 2 + 1)}_{\text{2nd row}} - \underbrace{(1 + 1 + 1)}_{\text{0th and 1st rows}} = 1. \) What is the minimum value of \(f(n)\) for \(n \ge 2015\)?

tertre  Mar 8, 2018
Sort: 

3+0 Answers

 #1
avatar+86604 
0

Here's the first, tertre

 

( x + y + z)^8  =   ( x  +  ( y + z) )^8  = 

 

C (8,5) * x^3 * (y + z)^5   .......we need to  find the coefficiient on y^3z^2  for the second part in red        

 

C (8,5) * x^3 * C(5,3)* y^3 * z^2  =

 

56* x^3  *  10 * y^3 * z^2   =

 

560   (x^3 * y^3 * z^2)

 

 

cool cool cool

CPhill  Mar 8, 2018
edited by CPhill  Mar 8, 2018
 #2
avatar
0

Just a minor typo in the 3rd to the last step. The second x^3 should read y^3.

Guest Mar 8, 2018
 #3
avatar+117 
+2

It will always be \(2^n-(2^{n-1}+2^{n-2} \cdots +2^1+2^0)=1\).Thus, the minimum value is \(\boxed{1}\)

azsun  Mar 8, 2018

26 Online Users

avatar
avatar
avatar
New Privacy Policy (May 2018)
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  Privacy Policy