Question

Coefficient of $t^{24} $ in $ (1+t^2)^{12}) (1+t^{12}(1+t^{24})$ is

• A. $^{12}C_6 +3$
• B. $^{12}C_6 +1$
• C. $^{12}C_6$
• D. $^{12}C_6 +2$

Answer 1

The Correct Option is D

Here, Coefficient of $t^{24}$ in $\{ (1+t^2)^{12}) (1+t^{12}(1+t^{24}) \}$
= Coefficient of $t^{24} $ in $ \{(1+t^2)^{12}
(1+t^{12}+t^{24}+ t^{36})\}$
= Coefficient of $t^{24} $ in
$ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \{(1+t^2)^{12}+t^{12}(1+t^2)^{12}+t^{24}(1+t^2)^{12} \};$
$\hspace27mm [ $ neglecting $ t^{36}(1 + t^2)^{12}]$
= Coefficient of $t^{24}\, =(^{12}C_{12}+^{12}C_6+^{12}C_0=2 +\, ^{12}C_6$

Answer 2

Ans. The algebraic expansion of the binomial (a+b) for a positive integral exponent n may be accomplished using the binomial theorem. When an expression's power rises, the computation is complicated and time-consuming. Therefore, even the coefficient of x²⁰ may be easily calculated using this theory. In the case of a random experiment, the theorem is crucial in establishing the probability of events. The exponent value of the binomial theorem expansion may be a fraction or a negative integer.

Binomial Theorem is the mathematical expression that consists of two terms including addition or subtraction operations. The equal terms should be combined to add the binomials and the distributive property must be used to multiply the binomials. For example, (1+x), (x+y), (x²+xy), and (2a+3b) are a few binomial expressions.

The coefficients in the binomial expansion of (a+b)ⁿ, n € N are called binomial coefficients. 

ⁿC₀, ⁿC₁, ⁿC₂ . . . . . . .ⁿC_n are some of the coefficients. Since ⁿC_r occurs as the coefficients of x^x in (1+x)ⁿ where n€N and as the coefficients of a^y. b^(n-y) in (a+b)ⁿ, they are called binomial coefficients.

These coefficient values of ⁿC_r can be arranged in the form of a triangle and are called the Pascal triangle. The (k+1) row consists of values ^kC₀, ^kC₁, ^kC₂, ^kC₃,…….,^kC_k