The prefix bi in the words bicycle, binocular, binary and in many more words means two.

**Binomial
Theorem**

The prefix bi in the words bicycle, binocular, binary and
in many more words means two. The word *binomial *stands for expressions
having two terms. For examples* *(1
+* **x*)*,** *(*x** *+* **y*)*,** *(*x*^{2}* *+* **xy*)* *and* *(2*a* + 3*b*) are some binomial
expressions.

**1. Binomial
Coefficients**

If we observe carefully the
Pascal triangle, we may notice that each row starts and ends with 1 and other
entries are the sum of the two numbers just above it. For example ‘3’ is the
sum of 1 and 2 above it; ‘10’ is the sum of 4 and 6 above it. We will prove in
a short while that

which is the *binomial expansion* of (*a*
+ *b*)^{n}. The binomial expansion of (*a* + *b*)^{n} for any *n* *∈* N can be written using
Pascal triangle. For example, from the fifth row we can write down the
expansion of (*a* + *b*)^{4}
and from the sixth row
we can write down the expansion of (*a* + *b*)^{5} and so on. We know the terms (without coefficients) of (*a*
+ *b*)^{5} are

The Pascal triangle
can be constructed using addition alone, without using any multiplication or
division. So without multiplication we can write down the binomial expansion
for (*a*
+ *b*)^{n} for any *n **∈** *N.

The above pattern
resembling a triangle, is credited in the name of the seventeenth century
French Mathematician Blaise Pascal, who studied mathematical properties of this
structure and used this concept effectively in Probability Theory.

**2. Binomial
theorem for positive integral index**

Now we prove the most
celebrated theorem called Binomial Theorem.

**Theorem 5.1 (Binomial theorem for positive integral index): ** If *n* is any positive integer, then

** **

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11th Mathematics : UNIT 5 : Binomial Theorem, Sequences and Series : Binomial Theorem | Binomial Coefficients | Definition, Formula, Solved Example Problems, Exercise | Mathematics