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Math  /  Discrete

QuestionA summer camp offers sessions in horseback riding (R)(\mathrm{R}), tennis (T)(\mathrm{T}), and sailing (S)(\mathrm{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: □\square (Simplify your answer.)

Studdy Solution

STEP 1

1. We are given the number of campers who signed up for various combinations of sessions.
2. We need to find the number of campers who signed up for both tennis and sailing but not horseback riding.

STEP 2

1. Understand the given information and set up the problem using set notation.
2. Use the principle of inclusion-exclusion to find the desired number of campers.
3. Calculate the number of campers who signed up for tennis and sailing but not horseback riding.

STEP 3

Define the sets: - R R : Campers who signed up for horseback riding. - T T : Campers who signed up for tennis. - S S : Campers who signed up for sailing.
Given: - n(R′)=23 n(R') = 23 - n(R∪S)=21 n(R \cup S) = 21 - n(S)=15 n(S) = 15 - n(T)=15 n(T) = 15 - n(R∩S)=6 n(R \cap S) = 6 - n(S∩T)=5 n(S \cap T) = 5 - n(R∩T)=6 n(R \cap T) = 6 - n(R∩S∩T)=3 n(R \cap S \cap T) = 3

STEP 4

We need to find n(S∩T∩R′) n(S \cap T \cap R') , which represents campers who signed up for tennis and sailing but not horseback riding.
Using the principle of inclusion-exclusion, we can express n(S∩T) n(S \cap T) as: n(S∩T)=n(S∩T∩R)+n(S∩T∩R′) n(S \cap T) = n(S \cap T \cap R) + n(S \cap T \cap R')
Given: n(S∩T)=5 n(S \cap T) = 5 n(S∩T∩R)=3 n(S \cap T \cap R) = 3

STEP 5

Substitute the known values into the equation: 5=3+n(S∩T∩R′) 5 = 3 + n(S \cap T \cap R')
Solve for n(S∩T∩R′) n(S \cap T \cap R') : n(S∩T∩R′)=5−3=2 n(S \cap T \cap R') = 5 - 3 = 2

STEP 6

The number of campers who signed up for tennis and sailing but not horseback riding is:
2 \boxed{2}

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