Let S = {1, 2, 3, ..., 12}. How many subsets of S, excluding the empty set, have an even sum but not an even product?

Step-by-step explanation:

The set S = {1, ,2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. The total number of elements in the set (n) is 12. The sum of the elements in the set S is even if the sum of the elements of the complement of the set S is odd.

The number of pairs that can give an even sum is therefore

Since the empty set is excluded, The number of pairs that can give an even sum is therefore

The answer is n=21.

Step-by-step explanation:

To solve the equation for (n), start by using the distributive property on both sides of the equation. The distributive property is multiplying the sum of two or more addends by a number which will give the same result as multiplying each addend individually by the number and then adding the products together.

The equation will look like after using the distributive property. Next, add 35 to both sides of the equation, which will look like , and then subtract (2n) from both sides of the equation, which will look like . The next step is to simplify the equation, which will look like , and the final answer to the equation will be .