Question

Determine the number of subsets of $\{1,2,3,4,............50\}$ whose sum of elements is larger than or equal to $638$.

• A. $2^{49}$
• B. $2^{48}$
• C. $2^{47}$
• D. $2^{46}$

Answer 1

The correct option is A $2^{49}$

Let $A=\{1,2,3,\mathrm{4......50}\}$

We know the sum of all elements in $A$ is $\frac{50\times 51}{2}=1275$

$A$ has $2^{50}$ subsets.

We group these subsets into pairs such that each pair adds up to $1275$

Hence there will be one set in each of the pairs which is greater than or equal to $638$ since $\frac{1275}{2}=637.5$

So total no. of required subsets = $\frac{2^{50}}{2}=2^{49}$