A survey of 500 adults found that
190 played golf, 200 skied, 95 played tennis, 100
played golf but did not ski or play tennis, 120 skied
but did not play golf or tennis, 30 played golf and
skied but did not play tennis, and 40 did all three.
a. How many played golf and tennis but did not
ski?
b. How many played tennis but did not play golf or
ski?
c. How many participated in at least one of the
three sports?

To solve this problem, we will use the principle of inclusion-exclusion and set notation to break down the information given in the survey. Let's define the sets:

- \( G \): Set of adults who played golf
- \( S \): Set of adults who skied
- \( T \): Set of adults who played tennis

### Given Information:

1. \( |G| = 190 \) (total who played golf)
2. \( |S| = 200 \) (total who skied)
3. \( |T| = 95 \) (total who played tennis)
4. \( |G \cap \overline{S} \cap \overline{T}| = 100 \) (played golf but did not ski or play tennis)
5. \( |S \cap \overline{G} \cap \overline{T}| = 120 \) (skied but did not play golf or tennis)
6. \( |G \cap S \cap \overline{T}| = 30 \) (played golf and skied but did not play tennis)
7. \( |G \cap S \cap T| = 40 \) (played all three)

### a. How many played golf and tennis but did not ski?

We need to find \( |G \cap T \cap \overline{S}| \), which is the number of adults who played both golf and tennis but did not ski.

The formula for the total number of people who played golf can be broken down as:

\[
|G| = |G \cap \overline{S} \cap \overline{T}| + |G \cap S \cap \overline{T}| + |G \cap \overline{S} \cap T| + |G \cap S \cap T|
\]

We know:
- \( |G \cap \overline{S} \cap \overline{T}| = 100 \)
- \( |G \cap S \cap \overline{T}| = 30 \)
- \( |G \cap S \cap T| = 40 \)

Substitute these values:

\[
190 = 100 + 30 + |G \cap \overline{S} \cap T| + 40
\]

Simplify to find \( |G \cap \overline{S} \cap T| \):

\[
190 = 170 + |G \cap \overline{S} \cap T|
\]

\[
|G \cap \overline{S} \cap T| = 20
\]

So, 20 adults played golf and tennis but did not ski.

### b. How many played tennis but did not play golf or ski?

We need to find \( |T \cap \overline{G} \cap \overline{S}| \), which is the number of adults who played tennis but did not play golf or ski.

The formula for the total number of people who played tennis can be broken down as:

\[
|T| = |T \cap \overline{G} \cap \overline{S}| + |G \cap \overline{S} \cap T| + |S \cap \overline{G} \cap T| + |G \cap S \cap T|
\]

We know:
- \( |G \cap \overline{S} \cap T| = 20 \)
- \( |G \cap S \cap T| = 40 \)

To find \( |S \cap \overline{G} \cap T| \), we apply the principle:

\[
|S \cap \overline{G} \cap T| = |S \cap T| - |G \cap S \cap T| = |S \cap T| - 40
\]

However, we first find \( |T \cap \overline{G} \cap S| = 95 \)

Total adults have golf, skiing, or tennis,

Now, skii, tennis, etc.

Finally,

Participants. 

Feel free back.

Explore a detailed solution to a set theory problem involving golf, skiing, and tennis. Learn how to apply the principle of inclusion-exclusion to find the number of adults who played golf and tennis but did not ski, those who played tennis but did not play golf or ski, and more.

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