SOLUTION: Hello, I'm at a loss of how to solve this. Any help would be very much appreciated. Thanks in advance! In Pascal's triangle, nCr+nCr+1=n+1Cr+1. Let X, Y, and Z be three consecut

Algebra.Com
Question 1083531: Hello, I'm at a loss of how to solve this. Any help would be very much appreciated. Thanks in advance!
In Pascal's triangle, nCr+nCr+1=n+1Cr+1. Let X, Y, and Z be three consecutive numbers in the nth row of Pascal's Triangle, such that X = nCr, Y = nCr+1, and Z = nCr+2. Use the nature of Pascal's Triangle to show that X + Y + Z = n+2Cr+2 - nCr+1.
Hint: Start by finding X + Y and Y + Z.

Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
To keep from getting bogged down with the n's and r's
Let's write the "Pascal's triangle formula" using
letters different from n and r:

sCt + sC(t+1) = (s+1)C(t+1)  <-- Pascal's triangle formula 

X = nCr, Y = nCr+1, and Z = nCr+2

X + Y = nCr + nC(r+1) = (n+1)C(r+1)

Y + Z = nC(r+1) + nC(r+2) 

Let s = n and t = r+1 in Pascal's triangle formula:

Y+Z = nC(r+1) + nC(r+1+1) = nC(r+1) + nC(r+2)

Adding the two equations together:

X + 2Y + Z = (n+1)C(r+1) + (n+1)C(r+2),

Now let s = n+1, and t = r+1 in Pascal's triangle formula:

(n+1)C(r+1) + (n+1)C(r+1+1) = (n+1+1)C(r+1+1) 

which simplifies to

(n+1)C(r+1) + (n+1)C(r+2) = (n+2)C(r+2) 

That left side turned out to be the same as the right side of 
X + 2Y + Z above. So

X + 2Y + Z = (n+1)C(r+1) + (n+1)C(r+2) = (n+2)C(r+2)

or

X + 2Y + Z = (n+2)C(r+2)

To get X + Y + Z we need to subtract Y = nC(r+1) from
both sides:

X + 2Y + Z = (n+2)C(r+2)
     Y     =   nC(r+1)
------------------------
X  + Y + Z = (n+2)C(r+2) - nC(r+1)

Edwin

RELATED QUESTIONS

Hello. I need some help with an irrational number problem. "Are irrational numbers... (answered by jim_thompson5910,stanbon)
Hello, I am needing some help in trying to figure out how to solve this equation:... (answered by tutorcecilia)
Hello! I've tried to approach this from a substiution standpoint but I'm at a loss :... (answered by venugopalramana)
This is a different kind of question any advice would be appreciated. I’m planning to go... (answered by stanbon,ikleyn)
Hello, I need to find g(x), the inverse function of f(x)=log_2(4x)-2. I would also like... (answered by Theo)
A right triangle consists of: leg a = 1 leg b = unknown hypotenuse= {{{sqrt(5)}}}... (answered by mananth)
Hi guys, f(x)=x^2+2x+15 Find all complex numbers for which f(x)=0 I'm at a loss... (answered by AnlytcPhil)
Hello, I need help finding the explicit formula for this problem. I'm fairly certain that (answered by math_helper)
Very stumped on how to do this problem... any help would be greatly appreciated!! In a (answered by nyc_function)