Question

There are several sets of different numbers that can be chosen from {0,1,2,3,4,5,6,7,8,9}.

a) How many of these sets contain any 2 numbers?
b) How many of these sets contain any 5 numbers?

Answer 1

The number of different sets containing any 2 numbers from the given set is 45, and the number of different sets containing any 5 numbers is 252, calculated by using combinations from the basic principles of combinatorics in mathematics.

Step-by-step explanation:

The student is asking about combinations of numbers from a set which pertains to the area of combinatorics, a topic within Mathematics.

Part A (Any 2 numbers)

To find the number of sets containing any 2 numbers from the given set, we use combinations since the order does not matter. Given the set {0,1,2,3,4,5,6,7,8,9} which has 10 elements, the number of ways to choose 2 numbers out of these 10 is calculated using the binomial coefficient which is 10 choose 2, denoted as 10C2.

10C2 = 10! / (2! * (10-2)!) = 45

Therefore, 45 different sets contain any 2 numbers from the set {0,1,2,3,4,5,6,7,8,9}.

Part B (Any 5 numbers)

Again, using combinations for selecting 5 out of 10 numbers: 10C5.

10C5 = 10! / (5! * (10-5)!) = 252

Thus, 252 different sets contain any 5 numbers from the initial set.