The total no. of combinations of 2n different things taken any one or more at a time is given by

 

2nC1+ 2nC2 + 2nC3+ 2nC4 +... +  2nC2n = 22n - 1 .......(1)

 

Similarly, the total no. of combinations of n different things taken any one or more at a time is given by

 

nC1+ nC2 + nC3+ nC4 +... +  nCn = 2n - 1 .......(1)

 

Further it is Given that

(2²ⁿ - 1)/(2ⁿ - 1) = 65/1

 

let x = 2ⁿ

 

therefore, (x² - 1)/(x - 1) = 65/1

 

or x + 1 = 65

 

or x = 64

 

or 2ⁿ = 64

 

or n = 6

I guess the problem is clear and understandable now