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Math  /  Discrete

QuestionA summer camp offers sessions in horseback riding (R), tennis (T), and sailing (S). The number of campers who signed up for each of the sessions are given below: n(R′)=23,n(R∪S)=21,n(S)=15,n(T)=15,n(R∩S)=6,n(S∩T)=5,n(R∩T)=6n\left(R^{\prime}\right)=23, n(R \cup S)=21, n(S)=15, n(T)=15, n(R \cap S)=6, n(S \cap T)=5, n(R \cap T)=6, n(R∩S∩T)=3n(R \cap S \cap T)=3
Determine how many campers signed up for tennis and sailing, but not horseback riding.
Campers signed up for tennis and sailing but not horseback riding: (Simplify your answer.) □\square

Studdy Solution

STEP 1

1. The sets R R , T T , and S S represent campers signed up for horseback riding, tennis, and sailing, respectively.
2. The problem involves finding the number of campers in the intersection of tennis and sailing, excluding horseback riding.

STEP 2

1. Use the principle of inclusion-exclusion to find the number of campers in the intersection of tennis and sailing.
2. Subtract the number of campers who are also signed up for horseback riding from the result of Step 1.

STEP 3

Use the principle of inclusion-exclusion to find the number of campers signed up for both tennis and sailing:
n(T∩S)=n(T)+n(S)−n(T∪S) n(T \cap S) = n(T) + n(S) - n(T \cup S)
We need to find n(T∪S) n(T \cup S) using the given data:
n(T∪S)=n(T)+n(S)−n(T∩S) n(T \cup S) = n(T) + n(S) - n(T \cap S)
Given: - n(T)=15 n(T) = 15 - n(S)=15 n(S) = 15 - n(S∩T)=5 n(S \cap T) = 5
Thus:
n(T∩S)=15+15−5=25 n(T \cap S) = 15 + 15 - 5 = 25

STEP 4

Subtract the number of campers who are also signed up for horseback riding from the number of campers signed up for both tennis and sailing:
n((T∩S)∩R′)=n(T∩S)−n(T∩S∩R) n((T \cap S) \cap R') = n(T \cap S) - n(T \cap S \cap R)
Given: - n(T∩S∩R)=3 n(T \cap S \cap R) = 3
Thus:
n((T∩S)∩R′)=5−3=2 n((T \cap S) \cap R') = 5 - 3 = 2
The number of campers signed up for tennis and sailing, but not horseback riding is:
2 \boxed{2}

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