Binomial theorem (or Binomial Expansion) gives us a way for finding any power of a binomial without multiplying at length.

Index

**History**

1. Special cases of the binomial theorem were known since at least the **4th century BC** when Greek mathematician Euclid mentioned the **special case of the binomial theorem **for exponent.

2. There is evidence that the **binomial theorem for cubes** was known by the **6th century AD** in India.

3. In 1544, **Michael Stifel** introduced the term “**binomial coefficient**” and showed how to use them to express (1 + a)^{n} in terms of (1+a)^{n-1}, via “Pascal’s triangle“.

**Theorem**

**Statement** (Binomial theorem formula)

According to the theorem, it is possible to expand any nonnegative power of (*x* + *y*) into a sum of the form

Where, \(n\geq 0\ and\ 0\leq k\leq n,\ {N\choose k}\) is a positive integer known as binomial coefficient.

**Binomial Theorem Proof **(by Induction)

The proof will be given by using the **Principle of mathematical induction** (PMI). This is done by first proving it for n=1, then assuming that it is true for n=k, we prove it for n=k.

Let P(n):

Now, for **n=1** we have

So, it’s true for **n=1**

Now, for n=2 we have

So, it’s true for **n=2**

**Now let n=k be true,** i.e.

Thus, it has been **proved that P(k+1) is true for all values of k**.

Therefore, by principle of mathematical induction, **P(n) is true for every positive integer n**.

**Applications**

Binomial Theorem plays an important role in many different fields of mathematics as well other areas. Some of these are

1. **Multiple-angle identities**– In complex numbers, the binomial theorem is combined with de Moivre’s formula to yield multiple-angle formulas for the Sine and Cosine. The expression can be expanded, and then the real and imaginary parts can be taken to yield formulas.

2. **Series for e**– The number is defined by the formula

3. **Probability**– This theorem is closely related to the probability mass function. **Binomial probability** refers to the **probability** of exactly x successes on n repeated trials in an experiment which has two possible outcomes

**Binomial Theorem Examples**

1. **Find an approximation of (0.99) ^{5}.**

Sol. We can solve this by just multiplying 0.99 five times but a smart way would be by using binomial theorem in a clever way.

On solving the above expression we get the answer as **0.9509900499.**

**2. How many terms are present in the expansion of (x + a) ^{100} + (x – a)^{100} after simplification.**

Sol.

So, there are 101 terms in these expansions, and **50 odd powered (negative terms in the second expansion) gets cancelled** leaving us with 51 terms.

So, the expansion will have 51 terms,

**FAQs**

**What is the meaning of**\({x\choose y}\)?\({x\choose y}\) is another abbreviation for ^{x}C_{y}**.**

(OR)**\( \frac{x!}{y!(x-y)!}\)**

**How do we express binomial expansion in terms of summation?**It can be expressed as \( {(x+y)}^n = \sum^n_{k=0}{x\choose y}x^ky^{n-k} \)

**What is the meaning of binomial?**Binomial is basically a polynomial having two terms or two monomials.

**What is the meaning of Binomial probability?**Binomial probability refers to the probability of an event ‘x’ which successes on ‘n’ repeated trials in an experiment with only two possible outcomes.

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