In the expansion of ( [3] 2 + 1 [3] 3 )^n, n N , if the ratio of 15^ at term from the beginning to the 15^ th term from the end is 1 6 , then the value of ^n C₃ is:

Step-by-step solution

aligned & T _ r +1 = ^ n C _ r (2^ 1 / 3 )^ n - r ( 1 3^ 1 / 3 )^ r & r =14 & ~T ₁₅= ^ n C ₁₄ (2^ 1 / 3 )^ n -14 ( 1 3^ 1 / 3 )¹⁴ aligned T ₁₅^ =15^ h term from last is ( n -13)^ h term from beginning. aligned & T ₁₅^ = ^ n C _ n -14 (2^ 1 / 3 )¹⁴ ( 1 3^ 1 / 3 )^ n -14 & ~T ₁₅ ~T ₁₅^ = ^ n C ₁₄ (2^ 1 / 3 )^ n -14 ( 1 3^ 1 / 3 )¹⁴ ^ n C _ n -14 (2^ 1 / 3 )¹⁴ ( 1 3^ 1 / 3 )^ n -14 = 1 6 aligned aligned & = (2^ 1 / 3 )^ n-28 (3^ 1 / 3 )^ n-28 = 1 6 & =6^ n-28 3 =6⁻¹ aligned So, ^n C ₃= ²⁵ C ₃=2300