Most Important Selected Qs for JEE AdvancedMathematicsBinomial Theorem
Paragraph Let (1+x+x^2 )²⁰=a₀+a₁ x+a₂ x^2+ +a₄₀ x⁴⁰ . Further S= _ r=0 ⁴⁰ a_r . Two coefficients are chosen from the coefficients a ₀, a ₁, , a ₄₀ and the probability that they are equal is p (considering no three coefficients are equal). Units digit of S is equal to b and a= 1-p 10 p . On the basis of above information, answer the following : Question The coefficient a ₃ is equal to -
Options
- A3.20! 18!
- B4.20! 18!
- C20! 17!3!
- D20! 18!
Correct answer
B. 4.20! 18!
Step-by-step solution
S=3²⁰ unit's digit of S is b=1 aligned & Also p = ²⁰ C ₁ ⁴¹ C ₂ = 1 41 & a =4 aligned The general term in the expansion of (1+x+x^2 )²⁰ is 20! r!s!t! 1^r x^s x^ 2 t where r+s+t=20 for coefficient of x^3, s+2 t=3 aligned & t=0, s=3, r=17 & or t=1, s=1, r=18 aligned coefficient of x ^3= 20! 17!3! + 20! 18!1! = 4.20! 18!