Test: Binomial Theorem For Positive Index - JEE MCQ

Since this binomial is to the power 8, there will be nine terms in the expansion.

n = 10

Middle term = (n/2) + 1
= (10/2) + 1
= 6th term

T(6) = T(5+1)
= ¹⁰C₅[(2x²)/3]⁵ [(3/2x²)]⁵
= ¹⁰C₅
= 252

The total number of terms in the binomial expansion of (a + b)ⁿ is n + 1, i.e. one more than the exponent n.

(1-x)^-6 
=> (1-x)^(-6/1)
It is in the form of (1-x)^(-p/q), p =6, q=1

(1-x)^(-p/q) = 1+p/1!(x/q)¹ + p(p+q)/2!(x/q)² + p(p+q)(p+2q)/3!(x/q)³ + p(p+q)(p+2q)(p+3q)/4!(x/q)⁴........

= 1+6/1!(x/1)¹ + 6(7)/2!(x/1)² + 6(7)(8)/3!(x/1)³ + 6(7)(8)(9)/4!(x/1)⁴ +.......................

So, coefficient of x⁵ is (6*7*8*9*10)/120
= 252

Correct Answer: d

Explanation:- The coefficient of terms (x+a)ⁿ equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficients.

ⁿC_r = ⁿC_{n – r}, r = 0,1,2,…,n.

Number of terms(n) = 10 
Middle term = (n/2) + 1

= (10/2) + 1
= 5 + 1
= 6th term

Coefficients of the rth and (r+4)th terms in the given expansion are C_r−120  and ²⁰C_r+3.
Here,C_r−120  = ²⁰C_r+3
⇒ r−1+r+3 = 20 
[∵ if ⁿC_x = ⁿC_y  ⇒ x = y or x+y = n]

⇒ r = 2 or 2r = 18
⇒ r = 9  

Here the number of terms can be calculated by:
= ((n+ 1) * (n+2)) /2
where, n =7

∴ Number of terms = 36

No. of terms is ⁿ⁺²C₂

If a and b are real numbers and n is a positive integer, then:
(a - b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ a^{(n – 1)} b¹ + ⁿC₂ a^{(n – 2)} b²+ ...... + ⁿC_r a^{(n – r)} b^{r+ ... + n}C_nbⁿ,

The general term or (r + 1)th term in the expansion is given by:
T_{r + 1} = (-1)C_r a^(n–r) b^r