Binomial Theorem

1. If n is a positive integer ${(a + b)}^{n}$ = $a^{n}$ + n $C_{1}$ $a^{n - 1}$ b + n $C_{2}$ $a^{n - 2} b^{2}$ +......... +n $C_{n}$

$b^{n}$
2. ${(a - b)}^{n}$ = $a^{n}$ - n $C_{1}$ $a^{n - 1}$ b + n $C_{2}$ $a^{n - 2} b^{2}$ - ......... + ${(- 1)}^{n}$ n $C_{n}$ $b^{n}$

3. $T_{r + 1}$ = n $C_{r}$ $a^{n - r}$ . $b^{r}$ in (1)

= ${(- 1)}^{r}$ n $C_{r}$ $a^{n - r}$ . $b^{r}$ in (2)

4. Binomial coefficients of terms equidistant from begining and end are equal.

ie, n $C_{0}$ = n $C_{n}$ , n $C_{1}$ = n $C_{n - 1}$ , n $C_{2}$ = n $C_{n - 2}$ and so on.

5. Number of terms in the expansion = ( $\overset{n - 2}{n} + 1)$

6. Middle term if n even: ${(\frac{n}{2} + 1)}^{th}$ term. If n is odd: $\frac{n + 1}{2}$ and $\frac{n + 3}{2}$ th terms.

7. ${(1 + x)}^{n}$ = $C_{0}$ + $C_{1}$ x + $C_{2} x^{2}$ + $C_{3} x^{3}$ +..........+ $C_{n} x^{n,}$ , then

1. $C_{0}$ + $C_{1}$ + $C_{2}$ +...... + $C_{n}$ = $2^{n}$ .

2. $C_{1}$ + $C_{2}$ + $C_{3}$ +...... + $C_{n}$ = $2^{n} - 1$

3. $C_{0}$ - $C_{1}$ + $C_{2}$ - $C_{3} +$ ...... = 0

4. $C_{0}$ + $C_{2}$ + $C_{4}$ +...... = $C_{1}$ + $C_{3}$ + $C_{5} + .......$ = $2^{n - 1}$

5. $C_{1}$ + ${2 C}_{2}$ + ${3 C}_{3}$ +...... +n $C_{n}$ = n. $2^{n - 1}$

6. $C_{1}$ - ${2 C}_{2}$ + ${3 C}_{3}$ -...... = 0

7. $C_{0}$ + $\frac{C_{1}}{2}$ + $\frac{C_{2}}{3}$ +......+ $\frac{C_{n}}{n + 1}$ = $\frac{2^{n + 1} - 1}{n + 1}$

8. $C_{0}^{2}$ + $C_{1}^{2}$ + $C_{2}^{2}$ + ........ + $C_{n}^{2}$ = ${}^{2 n}{C_{n}}$

9. $C_{0}$ $C_{1}$ + $C_{1}$ $C_{2}$ +.......+ $C_{n - 1}$ $C_{n}$ = ${}^{2 n}{C_{n - 2}}$

10. $C_{0}$ $C_{2}$ + $C_{1}$ $C_{3}$ +........+ $C_{n - 2}$ $C_{n}$ = ${}^{2 n}{C_{n - 2}}$

11. $C_{0}$ $C_{r}$ + $C_{1}$ $C_{r + 1}$ +........+ $C_{n - r}$ $C_{n}$ = ${}^{2 n}{C_{n - r}}$

8. To find the greatest term in ${(1 + x)}^{n}$ where x $>$ 0, n $\in$ N is $T_{m + 1}$ where m is the largest +ve integer 'r' satisfying $\frac{T_{r + 1}}{T_{r}}$ = $(\frac{n - r + 1}{r})$ x $\geq$ 1
If for r=m, the equality holds then both $T_{m}$ + $T_{m + 1}$ are greatest and are equal.

9. To get the absolute term : In $T_{r + 1}$ find r such that power of x = 0

10. Number of terms in ${(x + y + z)}^{n}$ = $\frac{(n + 1) (n + 2)}{2}$
11. If | x | $<$ 1

${(1 + x)}^{n}$ = 1+nx + $\frac{n (n - 1)}{2}$ $x^{2}$ + $\frac{n (n - 1) (n - 2)}{3!}$ $x^{3}$ +.......... $\infty$

12. ${(1 - x)}^{n}$ = 1- nx+ $\frac{n (n - 1)}{2!}$ $x^{2}$ - $\frac{n (n - 1) (n - 2)}{3!}$ $x^{3}$ +........ $\infty$

13. ${(1 - x)}^{- 1}$ = 1 + x + $x^{2}$ + $x^{3}$ +...... $\infty$

14. ${(1 + x)}^{- 1}$ = 1 - x + $x^{2}$ - $x^{3}$ + ...... $\infty$

15. ${(1 - x)}^{- 2}$ = 1 + 2x + ${3 x}^{2}$ + ${4 x}^{3}$ +...... $\infty$

16. ${(1 + x)}^{- 2}$ = 1- 2x + ${3 x}^{2}$ + ${4 x}^{3}$ +..... $\infty$

17. ${(1 - x)}^{- 3}$ = 1+3x+ $\frac{3.4}{1.2}$ $x^{2}$ + $\frac{4.5}{1.2}$ $x^{3}$ +...... $\infty$

18. ${(1 + x)}^{- 3}$ = 1-3x+ $\frac{3.4}{1.2}$ $x^{2}$ - $\frac{4.5}{1.2}$ $x^{3}$ +...... $\infty$

19. In the expansion of ${(x! + a)}^{n}$ , ${(m + 1)}^{th}$ term from the end= ${(n - m + 1)}^{th}$ term from the beginning= $T_{n - m + 1}$