The Venn diagram shows circles H and N intersecting within set U.

Question

The universal set U is the set of all 29 flavors sold at Mrs. Lee's Ice Cream Shop.

H is the set of 11 flavors that contain chocolate chips.
N is the set of 7 flavors that contain nuts.
There are 4 flavors of ice cream that contain ,begin underline,both,end underline, nuts and chocolate chips.
Question
How many flavors of ice cream contain ,begin emphasis,neither,end emphasis, nuts nor chocolate chips?

Answer options with 5 options
A.
3

B.
7

C.
11

D.
15

E.
25

Answer 1

To solve the problem, we can use the principle of inclusion-exclusion for the sets involved.

1. Identify the sets:

   • Let \( |H| \) be the number of flavors with chocolate chips = 11.
   • Let \( |N| \) be the number of flavors with nuts = 7.
   • Let \( |H \cap N| \) be the number of flavors with both chocolate chips and nuts = 4.

2. Use the inclusion-exclusion principle to find the number of flavors that have either chocolate chips or nuts or both: \[ |H \cup N| = |H| + |N| - |H \cap N| \] Plugging in the values: \[ |H \cup N| = 11 + 7 - 4 = 14 \]

3. Find the number of flavors that contain neither nuts nor chocolate chips: The total number of flavors in the universal set \( U \) is 29. Therefore, the flavors that contain neither are: \[ |U| - |H \cup N| = 29 - 14 = 15 \]

Thus, the answer is that 15 flavors of ice cream contain neither nuts nor chocolate chips.

The correct answer is: D. 15