How many elements are in the union of five sets if the sets contain 10,000 elements each, each pair of sets has 1000 common elements, each triple of sets has 100 common elements, every four of the sets have 10 common elements, and there is 1 element in all five sets?

40951

Step-by-step explanation:

Using the principles of inclusion - Exclusion

Where C (n, r) = n! / (n-r) ! r!

Total elements in the five sets including number repetition is given as (10000) * C (5, 1) = 10000 * 5! / (5-1) !1!=10000*5=50000

Total Number of elements in each pair including number repetition of sets is given as

= (1000) * C (5, 2) = 10000

Number of elements in each triple of sets is given as

= (100) * C (5, 3) = 1000

Number of elements in every four sets

= (10) * C (5, 4) = 50

Number of elements in every one set

(1) * C (5, 5) = 1

Therefore total number of unique elements=50000-10000+1000-50+1

=40951