## Binomial Theorem

# Binomial Theorem for Positive Integral Indices

**Tips :**

- If ‘n’ is a positive integer, then (x + a)
^{n}=^{n}C_{o}. x^{n}+^{n}C_{1}. x^{n-1}. a +^{n}C_{2}. x^{n-2}. a^{2}+ ....... +^{n}C_{n}. a^{n} ^{n}C_{o}=^{n}C_{n};^{n}C_{r}=^{n}C_{n-r}- If ‘n’ is odd, there are two greatest binomial coefficients in the expansion which are \tt ^n C _\frac{n-1}{2}\ and\ ^n C _\frac{n+1}{2}
- If ‘n’ is even there is only one greatest binomial coefficient in the expansion (x + a)
^{n}which is \tt ^n C _\frac{n}{2} - If n > 2, n ∈ N then (2n −1)
^{n}+ (2n)^{n}< (2n+1)^{n} - If the coefficients of X
^{r-1}, X^{r}, X^{r+1}in (1 + x)^{n}are in A.P then (n −2r)^{2}= n + 2

**Tricks :**

- Coefficient of X
^{p}Y^{q}Z^{r}in (ax + by + cz)^{n}is \tt \frac{n! a^{p}b^{q}c^{r}}{p! \times q! \times r!} where p+q+r = n and n, p, q, r ∈ N. - (x + a)
^{n}– (x - a)^{n}= 2 [^{n}C_{1}x^{n−1}a +^{n}C_{3}x^{n−3}a^{3}+^{n}C_{5}x^{n−5}a^{5}+ ...........]

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### Part4: View the Topic in this video From 00:10 To 11:16

### Part5: View the Topic in this video From 00:10 To 07:30

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**Binomial Theorem for Positive Integer**:

If n is any positive integer, then (x + a)^{n} = ^{n}C_{0 }x^{n} + ^{n}C_{1 }x^{n−1} + a+ ^{n}C_{2} x^{n}^{−2} a^{2} + ... + ^{n}C_{n} a^{n}.

i.e., (x + a)^n \sum_{r=0}^n \ ^nC_r\ x^{n-r}a^{r}

2. (x − a)^{n} = ^{n}C_{0} − ^{n}C_{1} x^{n}^{−1} a + ^{n}C_{2} x^{n}^{−2} a^{2} − ^{n}C_{3} x^{n}^{−3} a^{3} + ... + (−1)^{n}. ^{n}C_{n} a^{n}

i.e., (x-a)^n \sum_{r=0}^n \ \left(-1\right) \ ^nC_r \cdot x^{n-r}\cdot a^{r}

3. (1 + x)^{n} = ^{n}C_{0} + ^{n}C_{1} x + ^{n}C_{2} x^{2} + ... + ^{n}C_{n} x^{n}

i.e., (1+x)^n=\sum_{r=0}^n \ ^nC_r \cdot x^{r}

4. (x − a)^{n} = ^{n}C_{0} − ^{n}C_{1} x + ^{n}C_{2} x^{2} − ^{n}C_{3} x^{3} + ... + \left(-1\right)^{r^{n}}C_{r}x^{r} + ...+ (-1)^{n} ^{n}C_{n} x^{n}

i.e.,(1-x)^n \sum_{r=0}^n \left(-1\right)^r \ {^{n}C_{r}}\cdot x^{r}

5. (x + a)^{n} + (x − a)^{n} = 2 (^{n}C_{0}x^{n} a^{0} + ^{n}C_{2} x^{n}^{−2} a^{2} + ...)

(x + a)^{n} − (x − a)^{n} = 2 (^{n}C_{1}x^{n}^{−1} a + ^{n}C_{3} x^{n}^{−3} a^{2} + ...)