LET S B A SET WITH 100 ELEMENTS .HOW MANY SUBSETS DOES THE SET HAVE WHICH CONTAIN ATLEAST 50 ELEMENTS?

To determine how many subsets of a set with 100 elements contain at least 50 elements, we can use some combinatorial principles. Let's break this down step by step.

Understanding Subsets

A subset is any combination of elements from a set. For a set with \( n \) elements, the total number of subsets is given by \( 2^n \). This includes all possible combinations, from the empty set to the full set itself.

Calculating Total Subsets

For our set \( S \) with 100 elements, the total number of subsets is:

• Total subsets = \( 2^{100} \)

Focusing on Subsets with at Least 50 Elements

Now, we want to find the number of subsets that contain at least 50 elements. This means we need to consider subsets that have 50, 51, 52, and so on, up to 100 elements.

Using Combinations

The number of ways to choose \( k \) elements from a set of \( n \) elements is given by the binomial coefficient, denoted as \( C(n, k) \) or \( \binom{n}{k} \). For our case, we need to sum the combinations from 50 to 100:

• Number of subsets with exactly \( k \) elements = \( \binom{100}{k} \)

Summing the Combinations

Thus, the total number of subsets with at least 50 elements can be expressed as:

• Number of subsets with at least 50 elements = \( \sum_{k=50}^{100} \binom{100}{k} \)

Using the Binomial Theorem

To simplify this calculation, we can use a property of binomial coefficients. The sum of all subsets of a set can be expressed as:

• Sum of all subsets = \( \sum_{k=0}^{100} \binom{100}{k} = 2^{100} \)

Since the subsets can be divided into those with fewer than 50 elements and those with at least 50 elements, we can express this as:

• Number of subsets with fewer than 50 elements = \( \sum_{k=0}^{49} \binom{100}{k} \)
• Number of subsets with at least 50 elements = \( 2^{100} - \sum_{k=0}^{49} \binom{100}{k} \)

Final Calculation

Using the symmetry of binomial coefficients, we know that:

• \( \sum_{k=0}^{49} \binom{100}{k} = \sum_{k=51}^{100} \binom{100}{k} \)

Thus, we can conclude that:

• Number of subsets with at least 50 elements = \( \frac{2^{100}}{2} = 2^{99} \)

Conclusion

Therefore, the number of subsets of a set with 100 elements that contain at least 50 elements is \( 2^{99} \). This is a significant number, illustrating the vast possibilities of combinations even within a relatively small set of elements.