In the expansion ( 1 x +x x )¹⁰, the co - efficient of 6^ th term is equal to 7 7 8 , then the principal value of x is

Step-by-step solution

The general term T_ r+1 in the expansion of (a+b)^ n is given by ^ n C_ r a^ n-r b^ r . For the expansion ( 1 x + x x )¹⁰ , the 6^ th term corresponds to r = 5 . T₆ = ¹⁰C₅ ( 1 x )¹⁰⁻⁵ (x x)⁵ = ¹⁰C₅ 1 x⁵ x⁵ ⁵ x = ¹⁰C₅ ⁵ x . The coefficient of the 6^ th term is ¹⁰C₅ ⁵ x . Given that the coefficient is 7 7 8 = 63 8 , we have ¹⁰C₅ ⁵ x = 63 8 . Calculating ¹⁰C₅ = 10 9 8 7 6 5 4 3 2 1 = 252 . So, 252 ⁵ x = 63 8 . ⁵ x = 63 8 252 = 63 2016 = 1 32 . x = ( 1 32 )^ 1/5 = 1 2 . The principal value of x for x = 1 2 is x = 30^ .