Question

How many subsets of size two are there of {1,2,3...20} which don't consist of two consecutive integers?

a) 90
b) 171
c) 190
d) 180

Answer 1

To determine the subsets of size two from the set {1,2,3...20} without consecutive numbers, we subtract the 19 pairs of consecutive numbers from the total combinations of size two. The total number is 171 subsets.

Step-by-step explanation:

The question asks for the count of subsets of size two from the set {1,2,3...20} that do not include consecutive integers. To solve this, we can look at the total number of two-number combinations (without restrictions) and then subtract the number of combinations that do include consecutive integers.

• First, calculate the total number of two-number combinations from a set of 20 using the combination formula C(n, k) = n! / (k! * (n-k)!), where n is the total number of elements, and k is the size of the subset. In this case, it would be C(20, 2).
• Then, recognize that there are 19 pairs of consecutive numbers in the set {1,2,3...20}. Since each of these pairs cannot be included in the subsets, we have to subtract this from the total number of combinations.
• By doing the math, you will find that there are a certain number of non-consecutive subsets.

Calculating the exact number, we find that there are 171 subsets of size two that do not contain consecutive numbers.