Question

In: Advanced Math

Let S = {1,2,3,...,10}. a. Find the number of subsets of S that contain the number...

Let S = {1,2,3,...,10}.

a. Find the number of subsets of S that contain the number 5.

b. Find the number of subsets of S that contain neither 5 nor 6.

c. Find the number of subsets of S that contain both 5 and 6.

d. Find the number of subsets of S that contain no odd numbers.

e. Find the number of subsets of S that contain exactly three elements.

f. Find the number of subsets of S that contain exactly three elements, one of which is 3.

g. Find the number of subsets of S that contain exactly five elements, all of them even.

h. Find the number of subsets of S with exactly five elements, including 3 or 4 but not both.

Expert Solution

a) Number of subsets that contain 5 - For each number 1, 2, ... 10 (except 5) we have 2 possibilities - it contains that number or not. Total such subsets is

b) Don't neither 5 nor 6 is as for each of the rest 8 numbers we have 2 possibilities - it contains that number or not.

c) Containing both - for the rest 8 numbers we have 2 possibilities - it contains that number or not. Again the number of such subsets is

d) There are 5 odd numbers - 1, 3, 5, 7, 9 so that we can 5 even numbers. Total number of subsets is

e) Exactly 3 elements is

f) 3 is always there. We have to choose 2 numbers from remaining 9 numbers so that is the number

g) There are exactly 5 even numbers in S. Thus there is only one subset of 5 elements with all of them even namely

h) Number of subsets with exactly 5 elements is

Number of subsets with 5 elements containing 3 is

Number of subsets with 5 elements containing 4 is

Number of subsets with 5 elements containing 3 and 4 is

Required number of subsets is

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Colby Messinger answered 2 years ago

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