Binomial Theorem
Coefficient comparison in binomial expansion
nta_pyq_2025_apr
Grade 12
Question:
in the expansion of ($\sqrt{2}$+ 3 1 ) ,n$\ in $N , if the ratio of 15 at term from the beginning to the 15 th term from the 3$\sqrt{3}$end is 1 6 , then the value of n$C_{3}$is:
$4060$
$1040$
$2300$
$4960$
Step-by-Step Solution
Key Concept: Use the general term of the binomial expansion for coefficient comparison$i_n$binomial expansion and impose the required coefficient or exponent condition.
n$1/3$$n-r$1$Tr+1 = Cr$(2 ) ( ) (3)$1/3$3$r = 14$14$n-14$1 n$1/3$$T_{15}$=$C_{14}$(2 ) ( )$1/3$3 T ′$15 = 15$h term from last is$(n - 13)$term from beginning. h$n-14$14 ′ 1 n$1/3$T =$C_n$-14 (2 ) ( ) 15$1/3$3$n-14$14 n$1/3$1$C_{14}$(2 ) ( )$T_{15}$3$1/3$1 $\Rightarrow$ = = ′ 14$n-14$$T_{15}$1 6 n$1/3$$C_n$-14 (2 ) ( )$1/3$3$n-28$$n-28$1$1/3$$1/3 = (2$) (3$) = 6$$n-28 -1 = 6$$3 = 6$n 25 So,$C_{3}$=$C_{3}$= 2300
Correct Answer: 3