Question

If a given set has ten elements, how many of its subsets the given number of elements?

(a) At most two elements.
(b) At least eight elements.
(c) More than two elements.
(d) From three to seven elements.

Answer 1

To determine the number of subsets with a given number of elements in a set of 10 elements, you can use the formula 2^n, where n is the number of elements in the set. For different scenarios, there are 56, 1013, and 912 subsets with the given number of elements.

Step-by-step explanation:

To determine the number of subsets with a given number of elements, we can use the formula 2^n, where n is the number of elements in the set. For (a) At most two elements, there are 10 choose 0 + 10 choose 1 + 10 choose 2 subsets, which is 1 + 10 + 45 = 56 subsets. For (b) At least eight elements, there are 10 choose 8 + 10 choose 9 + 10 choose 10 subsets, which is 45 + 10 + 1 = 56 subsets. For (c) More than two elements, there are 2^n - 10 choose 0 - 10 choose 1 subsets, which is 2^10 - 1 - 10 = 1013 subsets. For (d) From three to seven elements, there are 10 choose 3 + 10 choose 4 + 10 choose 5 + 10 choose 6 + 10 choose 7 subsets, which is 120 + 210 + 252 + 210 + 120 = 912 subsets.