**Binomial Co-efficients**

**Σ ^{n}C_{k }=
1 (k=0)**

**Σ ^{n}C_{k }=
1 + n (k=0 to k=1)**

**Σ ^{n}C_{k }=
1 + n(n+1)/2 (k=0 to k=2)**

**Σ ^{n}C_{k }=
1 + n(n+1)/2 +( n(n-1)(n-2)/2) - Σ_{n-->3
to n}[(n-1)(n-2)
+ (n-2)(n-3) +...... (n-n)(n-n+1)] (k=0 to
k=3)**

**Σ ^{n}C_{k }
= 1 + n(n+1)/2 + [n(n-1)(n-2)/2] - Σ_{n-->3
to n}[(n-1)(n-2)
+ (n-2)(n-3) +...... (n-n)(n-n+1)] +1/2 [1*(n-2)(n-3)+2*(n-3)(n-4)
+**

**
3*(n-4)(n-5) + .... +(n-3)*(n-(n-2))(n-(n-1))
] (k=0 to k=4
and (n-3) is the last
term co-efficient)**

** Q _{k } is
the ratio of frequency for k and frequency for k-1 i.e Q_{k}
=((n+1-k)/k)*(x/y) =q_{k}
(x/y)**

**where q _{k }=
(n+1-k)/k**

**Property of q _{k
and }Q_{k }
(excluding behavior at
0≤k<1**)

_{}***Q _{k }
** monotonically decreases as k increases, n remaining constant.

** Q _{k }
**(minimum) =(1/n)*

***q**** _{k }
** monotonically decreases as k increases, n remaining constant.

**
q _{k }
**(minimum) =(1/n)

** q _{k }
** =1 when k=(n+1)/2 i.e. when n is an

**Q**** _{k }
** =1 when k=(n+1)/(z+1) where z = (y/x) and k has
to be an integer.

* If we define a function f (x) such that Y = f (x) = (n+1-x)/x (do not confuse Y with small y) , then

Y =(n+1)/x - 1. Define another function z
=f (Y) = Y + 1 , then ** z =f _{1}(x) = (n+1)/x.
**Since n+1 is a constant say A,

* This is the Equation of a Equilateral or
Rectangular Hyperbola with asymptotes taken
as the co-ordinate axis and is of the form xy=a^{2}/2 = constant.* The
coordinates of the vertices are on the bisector of the first and third quadrant
and the first and second coordinate coincide, that is to say, x = z. Also, Point
A belongs to the curve of the hyperbola.

**Vertices lie in 1st,3rd
quadrant and are (√A,√A) and (-√A,-√A)**

Length of each Semi-Axis (here a=b) a, b is **√(2A). Length of Semi -axis c is √(a ^{2}+b^{2})
=√(a^{2}+a^{2}) = √(2a^{2}) =√(2.2A) =2√A**

Focus is **(√(2A),√(2A) ) , (-√(2A),-√(2A) ). Eccentricity=c/a= √2
(Eccentricity of hyperbola is always greater than 1)**