10 5 points A B and C are subsets of a universel set U such

10. (5 points) A, B, and C are subsets of a universel set U such that n(U) = 230, n(A) = 58, n(B) = 35, n(C) = 34, B C = theta , and n(( AU B U C)^?) = 130. How many members of U are in exactly one of the sets A, B and C? (A) 27 (B) 127 (C) 66 (D) 130 (E) 73 (F) 31 (G) 42 (B) 65 (J) 230 (K) insufficient information (L) none of the others

Solution

Here, we find the union of the three sets first,

n(A U B U C) =n(U) - n(A U B U C)\' = 230 - 130

n(A U B U C) = 100

Now,

n(B n C) = 0

Thus,

n(A n B n C) = 0

Now, note that

n(A U B U C) = n(A) + n(B) + n(C) - n(A n B) - n(A n C) - n(B n C) + n (A n B n C)

Thus,

100 = 58 + 35 + 34 - [n(A n B) + n(A n C) + 0] + (0)

100 = 127 - [n(A n B) + n(A n C)]

n(A n B) + n(A n C) = 27

Thus, those in exactly 1 set is

n(A only or B only or C only) = n(A U B U C) - [n(A n B) + n(A n C)]

= 100 - 27 = 73 [ANSWER, OPTION E]