Y is a subset of A such that the geometric mean of no two elements of Y is 350.

Question

Given  A = { 1, 3, 9, 27, 81,................................3¹⁰⁰ }

Y is a subset of A such that the geometric mean of no two elements of Y is 3⁵⁰. N is the maximum possible number of elements in set Y.

A. What is the value of N?

1. 50
2. 51
3. 49
4. 52

B. In how many ways can the set Y be formed such that it has exactly N elements?

1. 2⁵⁰
2. 2⁵¹
3. 3⁵⁰
4. 3⁵¹

Answer 1

Geometric mean of two numbers is 3⁵⁰ if their multiple is 3¹⁰⁰. This happens for 3^x and 3^y, where x + y = 100. So, out of 101 elements in A, we can have 101 possible values for (x, y) which means 50 pair of values. For these 50 pairs, we can only pick 1 value to be in Y. So, maximum number of elements in Y is 51. 

Out of 51 elements we have a choice out of 2 for 50 elements. So, no. of ways in which set Y can be formed = 2⁵⁰.