Question

Let \[S=\left\{ 1,2,3,...,100 \right\}\]. Determine the number of non-empty subsets \[A\] of the set \[S\] such that the product of elements in A is even.
(a) \[{{2}^{50}}\left( {{2}^{50}}-1 \right)\]
(b) \[{{2}^{100}}-1\]
(c) \[{{2}^{50}}-1\]
(d) \[{{2}^{50}}+1\]

Answer 1

Hint: In this question, we are given that \[S=\left\{ 1,2,3,...,100 \right\}\] where the number of elements in the set \[S\] is 100. Now we know that the sequence of first 100 natural numbers there are 50 odd natural numbers and 50 even natural numbers. Also we know that for a set \[X\] containing \[n\] elements, the total number of subsets of \[X\] is given by \[{{2}^{n}}\] which is also known as the power set of set \[X\]. Now we are given a set with 100 elements. So the total number of subsets of \[S\] is given by \[{{2}^{100}}\]. Since there are 50 odd natural numbers in the set \[S\], therefore the number of subsets of set \[S\] such that the product of elements is odd is given by \[{{2}^{50}}\] since we know that only product of two odd numbers is odd. Otherwise the product of an even and an odd is even number and product of two even number is also even number. Now in order to determine the number of non-empty subsets \[A\] of the set \[S\] such that the product of elements in \[A\] is even we will subtract the number of subsets of set \[S\] such that the product of elements in \[A\] is odd from the total number of subsets of set \[S\].

Complete step-by-step answer:
We are given that \[S=\left\{ 1,2,3,...,100 \right\}\] where the number of elements in the set \[S\] is 100. Since we know that for a set \[X\] containing \[n\] elements, the total number of subsets of \[X\] is given by \[{{2}^{n}}\] which is also known as the power set of set \[X\] denoted by \[P\left( X \right)\].
Therefore for the set \[S=\left\{ 1,2,3,...,100 \right\}\] where the number of elements in the set \[S\] is 100, the total number of subsets of set \[S\] is given by
\[{{2}^{100}}\]
We also know that the sequence of first 100 natural numbers there are 50 odd natural numbers and 50 even natural numbers.
Since the product of two odd numbers is odd and the product of an even and an odd is even number and product of two even numbers is also even number.
So in order to form subsets of set \[S\] such that the product of elements in the subset is odd, for that we have to have only odd numbers in the subset.
Since there are only 50 odd natural numbers in the set \[S\], therefore the total number of subsets of set \[S\] such that the product of elements in the subset is odd is given by
\[{{2}^{50}}\]
Now in order to determine the number of non-empty subsets \[A\] of the set \[S\] such that the product of elements in \[A\] is even we will subtract the number of subsets of set \[S\] such that the product of elements in \[A\] is odd from the total number of subsets of set \[S\].
Therefore the number of non-empty subsets \[A\] of the set \[S\] such that the product of elements in \[A\] is even is given by
\[\begin{align}
  & {{2}^{100}}-{{2}^{50}}={{\left( {{2}^{50}} \right)}^{2}}-{{2}^{50}} \\
 & ={{2}^{50}}\left( {{2}^{50}}-1 \right)
\end{align}\]
Hence the number of non-empty subsets \[A\] of the set \[S\] such that the product of elements in \[A\] is equals to \[{{2}^{50}}\left( {{2}^{50}}-1 \right)\].

So, the correct answer is “Option A”.

Note: In this problem, we can to determine the number of non-empty subsets \[A\] of the set \[S\] such that the product of elements in \[A\] is even we will subtract the number of subsets of set \[S\] such that the product of elements in \[A\] is odd from the total number of subsets of set \[S\]. Also take care of the fact that in the sequence of first 100 natural numbers there are 50 odd natural numbers and 50 even natural numbers. The product of two odd numbers is odd and the product of an even and an odd is even number and product of two even numbers is also even number.