Binomial Expansion

The expansion is of the type **(x + y )**^{n }where
x, y are two real numbers & n is an integer.

We have **(x + y )**^{n }= ^{ n}C_{0}x^{n}**y ^{0 }
**+

Or **(x + y )**^{n }=
**Σ _{k=0 --> }**

**1.** There are n+1 terms
in the expansion. If n is even , no. of terms is odd & vice versa. In case of
even terms, all coefficients occur in pairs. In case of odd terms , only the
middle coefficient does not occur in pair.

**2.** Arranging co-efficients
in rows for successive values of k from k=0 to k =n gives rise to Pascal's
triangle .

**3.** ** ^{n}C_{k }**= n! / [(k!)*(n-k)!] .
Therefore

**4**. ** ^{n}C_{k }**
=

**5. ** ** ^{n}C_{k }**
=

The expansion ( x + y )

The Coefficients form what is known as Pascal's Triangle. The characteristics are:-

1) No. of total terms is n+1. It starts with n=0 and ends with n=n.

2) Coefficients of n=0 and n=1 are 1.

3) The coefficients of any row can be obtained by adding the 2 numbers in the preceding row that are just at the left and just at the right of the given number.

4) Coefficients that are equidistant from the beginning and the end are equal

5) nC

6) to view the triangle click below

7) workout on factorials 4

8) binomial co-efficient chart

9) binomial calculation click

The general formula is:

put the value of n,k above, press calculate and then press CLICK below

**Probability Mass Function**

Probability distribution of a *
random discrete variable *is called probability mass function
whereas probability distribution of a

If k is a random discrete variable and there are n number of trials, and there are only 2 possible outcomes- success (probability-p) & failure (probability-q=1-p), then probability of getting k no. of successes in n number of trials is as per Binomial Distribution --

**P(p)= ^{ n}C_{k }p^{k
}q^{n-k}**

**
sum of the values of frequency at point k=0,1,2,.....n is equal to (p+q) ^{n}**

**B(k) / B(k-1) =((n+1-k)/k)*p/q =Q**

**Q _{k} > 1 when (n+1)p >
k(p+q) i.e frequency function value at that k is greater than its
previous k value.**

**Q _{k} = 1 when (n+1)p =
k(p+q) i.e frequency function value at that k is
equal to its previous k value.**

**Q _{k} < 1 when (n+1)p <
k(p+q) i.e frequency function value at that k is less than its previous k
value.**

**Most Probable Frequency : Supposing k has values n _{i }( n_{0},n_{1},n_{2},.......n_{n
, })**then if for any

**Uniform Sample Space:** If we toss 2 coins,
the sample points are (head ,head),(head,tail),(tail,head),(tail, tail). Each of
the 4 sample points have the same probability i.e 1/4. Hence sample space is
uniform. A sample space is said to be uniform if each point has the same
probability.

**Non-Uniform Sample Space: **A
sample space is said to be non-uniform if each point in it does not have the
same probability. For example, no. of heads in tossing of 4 coins. These are
** ^{n}C_{0}** ,

**For large value of independent trials
n: **When number of independent trials** n -->infinity, **binomial
distribution function tends to be the normal distribution function.

**Stirling's approximation:**
click