# Binomial Theorem Class 11

Binomial theorem class 11 – The binomial theorem states a formula for expressing the powers of sums. The most succinct version of this formula is shown immediately below.

$(x+y)^{r}\;=\;\sum_{k\;=\;0}^{\infty }\;\left ( \frac{r}{k} \right )X^{r-k}Y^{k}$

On simplifying the above equation we get:

$[a + b]^{n} = [ C_{n}^{0} \times a_{n} ] + [ C_{n}^{1} \times (a_{n-1}) \times b ] + [ C_{n}^{2} \times (a_{n-2}) \times b_{2} ] + . . . . . . . . . . . . . + [ C_{n}^{n-1} \times a \times (b_{n-1} ) ] + [ C_{n}^{n}\times b_{n} ]$

Some conclusions from Binomial Theorem:

(i) $[x + y]^{n} = [ C_{n}^{0} \times x^{n} ] + [ C_{n}^{1} \times (x^{n-1}) \times y ] + [ C_{n}^{2} \times (x^{n-2}) \times y^{2} ] + . . . . . . . . . . + [ C_{n}^{n-1} \times x \times (y^{n} – 1) ] + [ C_{n}^{n}\times y^{n} ]$

(ii) $[x – y]^{n} = [ C_{n}^{0} \times x^{n} ] – [ C_{n}^{1} \times (x^{n-1}) \times y ] + [ C_{n}^{2} \times (x^{n-2}) \times y^{2} ] – . . . . . . . . . . (-1)^{n}[ C_{n}^{n}\times y^{n} ]$

(iii) $[1 – x]^{n} = [C_{n}^{0}] – [ C_{n}^{1} \times x] + [ C_{n}^{2} \times x^{2}] – . . . . . . . . . . (-1)^{n}[ C_{n}^{n}\times x^{n} ]$

(v). $C_{n}^{0} = C_{n}^{n} = 1$

(vi). There are total (n + 1) terms in the expansion of $(a + b)^{n}$<

Binomial Theorem Class 11 Examples

#### Practise This Question

If  A, B, C are represented by 3 + 4i, 5 - 2i , -1 + 16i, then A, B, C are
[RPET 1986]