Set ` A,B,C, A cap B, A cap B, Acap C, B cap C and A cap B cap C ` have 35, 40, 45, 13, 12, 14 and 5 elements respectively. An element is selected at random from the set `a cup V cup C` . The probability that the selected element belongs to only set A is

Options

A. `(13)/(86)`
B. `(35)/(86)`
C. `(5)/(86)`
D. `(15)/(86)`

Answer

To solve the problem step-by-step, we will follow the instructions given in the video transcript and apply the inclusion-exclusion principle to find the required probability.

### Step 1: Identify the cardinalities of the sets
We are given the following cardinalities:
- \( N(A) = 35 \)
- \( N(B) = 40 \)
- \( N(C) = 45 \)
- \( N(A \cap B) = 13 \)
- \( N(A \cap C) = 12 \)
- \( N(B \cap C) = 14 \)
- \( N(A \cap B \cap C) = 5 \)

### Step 2: Apply the inclusion-exclusion principle
To find the total number of elements in the union of the three sets \( A \), \( B \), and \( C \) (denoted as \( N(A \cup B \cup C) \)), we use the inclusion-exclusion principle:

\[ N(A \cup B \cup C) = N(A) + N(B) + N(C) - N(A \cap B) - N(A \cap C) - N(B \cap C) + N(A \cap B \cap C) \]

### Step 3: Substitute the values into the formula
Now, we substitute the values we have:

\[ N(A \cup B \cup C) = 35 + 40 + 45 - 13 - 12 - 14 + 5 \]

### Step 4: Calculate the total
Now, we perform the arithmetic:

\[ N(A \cup B \cup C) = 35 + 40 + 45 = 120 \]
\[ 120 - 13 - 12 - 14 + 5 = 120 - 39 + 5 = 120 - 34 = 86 \]

Thus, we find that:

\[ N(A \cup B \cup C) = 86 \]

### Step 5: Find the number of elements in only set A
To find the number of elements that belong only to set \( A \) (denoted as \( N(A \text{ only}) \)), we can use the following formula:

\[ N(A \text{ only}) = N(A) - N(A \cap B) - N(A \cap C) + N(A \cap B \cap C) \]

### Step 6: Substitute the values
Substituting the values we have:

\[ N(A \text{ only}) = 35 - 13 - 12 + 5 \]

### Step 7: Calculate the number of elements in only set A
Now, we perform the arithmetic:

\[ N(A \text{ only}) = 35 - 13 - 12 + 5 = 35 - 25 = 10 \]

### Step 8: Calculate the probability
The probability that a randomly selected element belongs to only set \( A \) is given by:

\[ P(A \text{ only}) = \frac{N(A \text{ only})}{N(A \cup B \cup C)} = \frac{10}{86} \]

### Final Answer
Thus, the probability that the selected element belongs to only set \( A \) is:

\[ \frac{10}{86} = \frac{5}{43} \]