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
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