Solved Examples on Binomial Theorem

Illustration 1: In the binomial expansion of (a-b)ⁿ, n ≥ 5the sum of the 5^th and 6^th terms is zero. Then what is the value of a/b? (2001)

Solution: Let us denote the fifth term as T₅ and the sixth term as T₆.

So, it is given that T₅ + T₆  = 0

This gives ⁿC₄a^n-4 b⁴ - ⁿC₅a^n-5 b⁵ = 0

Hence, ⁿC₄a^n-4 b⁴ = ⁿC₅a^n-5 b⁵

In order to obtain the value of a/b, we shift the terms obtained above on one side.

This gives the value of a/b as = ⁿC₅/ ⁿC₄ = (n-4)/5.

Illustration 2:

Let T_n denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If T_n+1- T_n = 21, then what is the value of n? (2001)

Solution: According to the given condition, T_n = ⁿC₃

T_n+1- T_n = 21

Hence,ⁿ⁺¹C₃- ⁿC₃ = 21

So, (n+1)(n-1)n/6 – n(n-1)(n-2)/6 = 21

So, n(n-1)3/6 = 21

Hence, n(n-1) = 42

This gives, n =7.

Illustration 3: Coefficient of t²⁴ in (1 + t²)¹²(1 + t¹²)(1 + t²⁴) is….? (2003)

Solution: Here, coefficient of t²⁴ in (1 + t²)¹²(1 + t¹²)(1 + t²⁴)

This is same as the coefficient of t²⁴ in (1 + t²)¹²(1 + t¹² + t²⁴ + t³⁶)

Or the coefficient of t²⁴ in (1 + t²)¹²+t¹² (1 + t²)¹²+ t²⁴ (1 + t²)¹²

Hence,the coefficient of t²⁴ in ¹²C₁₂ + ¹²C₆ + ¹²C₀ = 2 + ¹²C₆