Interesting property of combinations

[Trenters4325]

How can you explain the fact that when you alternate (subtract, add, subtract, add, subtract, add...) any sequence of the form {nC0,nC1,nC2...nCn}, the corresponding series will always equal 0.

 

[galactus]

Notice the rows of Pascal's triangle. There is symmetry.

For instance, take the fifth row:

1 5 10 10 5 1

-1+5-10+10-5+1=0

See?.

 

[Trenters4325]

Symmetry only explains it when n is an odd number.

 

[galactus]

Try an even number then

The 6th row:

-1+6-15+20-15+6-1=0

It doesn't matter if it's odd or even.

 

[pka]

Come on both of you:
\(\displaystyle \L
\left( {x + y} \right)^n = \sum\limits_{}^{} {\left( \begin{array}{l}
n \\
k \\
\end{array} \right)x^{n - k} y^k }\).
Now let \(\displaystyle \L
x = 1\quad \& \quad y = - 1\).

 

[Trenters4325]

Nicely done.