Binomial Theorem

${\left(1+x\right)}^{11}={}^{11}{C}_0+{}^{11}{C}_{1}x+{}^{11}{C}_2{x}^2+.....{+}^{11}{C}_{11}{x}^{11}$

$=\frac{2^{12}-2-24}{12}$

$=\frac{2^{12}-26}{12}=\frac{4070}{12}=\frac{2035}{6}=\frac{m}{n}$

m + n = 2035 + 6 = 2041

$T_{n+1}={}^n{C}_{r}11^{\frac{1}{6}}\cdot 7^{\frac{824-4}{2}}$

For integral term

6 should divide r

and  $\frac{824-r}{2}$  must be integer

->2 most divide r

->r divisible by 6

->possible values of r Î {0, 1, 2, …824}

->For integer terms

r Î {0, 6, 12, …822} (822 = 0 + (n – 1)6 Þ n = 138)

= 138 terms

The common mistakes can be - using the formula incorrectly, before applying the formula students not simplify the expression, in Binomial Expansion, sometimes they forget to include the fractional and negative exponents, and using incorrect values for a, b, and n.

The Binomial model is known as the Lattice Model or Binomial Option Pricing Model.

There are a total of 36 questions comprising 14 straightforward formula-based sums, 16 challenging problems, and 6 intermediate.

Mathematically, binomial theorem formula is: (a + b)n = nC0 an + nC1 an - 1b + nC2 a n – 2b 2 + . + nCn – 1a.bn - 1 + nCn bn. Here a and b are real numbers and n is a positive integer.

An expression with two terms raised to a positive integer power can be expanded in Maths, such as (a + b)? This algebraic rule is known as the Binomial Theorem. 

But maximum value of k is not defined bonus

Kindly consider the following figure