Determine the number of subsets of A={1,2,…,10} whose sum of elements are greater than or equal to 28.

512

To determine the number of subsets of set A={1,2,…,10} whose sum of elements is greater than or equal to 28, we can use the concept of generating functions.

First, let's consider the generating function for the set A. The generating function for a set with elements {a1, a2, ..., an} can be represented as:

G(x) = (1 + x^a1)(1 + x^a2)...(1 + x^an)

In this case, the generating function for set A={1,2,…,10} would be:

G(x) = (1 + x^1)(1 + x^2)...(1 + x^10)

Now, we need to focus on the term that represents the sum of elements greater than or equal to 28. We can rewrite this as (x^28)(x^1 + x^2 + ... + x^10), where (x^1 + x^2 + ... + x^10) represents the sum of all possible elements in set A.

To find the coefficient in front of (x^28), we can multiply the coefficient of (x^28) in the product (x^1 + x^2 + ... + x^10) with the coefficient of (x^0) in the product (x^28).

To do this, we can expand the generating function G(x) as:

G(x) = (1 + x^1)(1 + x^2)...(1 + x^10)

After expanding, we can extract the coefficient in front of (x^28) in the term (x^28)(x^1 + x^2 + ... + x^10) to get the desired answer.

To simplify the explanation, let's work symbolically instead of working with actual numbers.

The coefficient in front of (x^1) in (1 + x^1) is 1.
The coefficient in front of (x^2) in (1 + x^2) is 1.
Similarly, the coefficients in front of (x^3), (x^4), ..., (x^10) are also 1.

Therefore, the product (x^1 + x^2 + ... + x^10) can be represented as:

(x^1 + x^2 + x^3 + x^4 + ... + x^10)

Now, let's find the coefficient in front of (x^28). We want to find the term that has (x^0) in the product (x^28). This is found by taking (x^28) raised to a negative power, which gives us (x^(-28)). In other words, we need to find the term with exponent -28.

To find the coefficient in front of (x^(-28)), we can look for the term that has (x^(-28)) in the expanded product, which is:

(x^10)(x^(-28))

The coefficient in front of (x^10) is 1, so the coefficient in front of (x^(-28)) is also 1.

Therefore, the number of subsets of set A={1,2,…,10} whose sum of elements is greater than or equal to 28 is 1.

Please note that this explanation may seem complicated due to the use of generating functions and working symbolically. In practice, using generating functions is a powerful technique to solve such problems by leveraging properties of polynomials and their exponents.