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

Binomial Theorem Class 11
Binomial Theorem Class 11
Binomial Theorem Class 11
Binomial Theorem Class 11
Binomial Theorem Class 11

 


Practise This Question

When three lines intersect at three points, then how many angles will be formed?