Sets A,B,C `AnnB,AnnC,BnnC` and `AnnBnnC` have 35, 40, 45, 13, 12, 14 and 5 elements respectively. An element is selected at random from the set A`uu`B`uu`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 need to find the probability that a randomly selected element from the set \( A \cup B \cup C \) belongs to only set \( A \).

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

### Step 2: Calculate the total number of elements in \( A \cup B \cup C \)
Using the principle of inclusion-exclusion, the formula for the total number of elements in \( A \cup B \cup C \) is:

\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \]

Substituting the values we have:

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

Calculating this step-by-step:

1. \( 35 + 40 + 45 = 120 \)
2. \( 120 - 13 = 107 \)
3. \( 107 - 12 = 95 \)
4. \( 95 - 14 = 81 \)
5. \( 81 + 5 = 86 \)

Thus,

\[ |A \cup B \cup C| = 86 \]

### Step 3: Calculate the number of elements that belong only to set \( A \)
To find the number of elements that belong only to set \( A \), we can use the following formula:

\[ |A \text{ only}| = |A| - |A \cap B| - |A \cap C| + |A \cap B \cap C| \]

Substituting the values:

\[ |A \text{ only}| = 35 - 13 - 12 + 5 \]

Calculating this step-by-step:

1. \( 35 - 13 = 22 \)
2. \( 22 - 12 = 10 \)
3. \( 10 + 5 = 15 \)

Thus,

\[ |A \text{ only}| = 15 \]

### Step 4: Calculate the probability
The probability \( P \) that a randomly selected element belongs to only set \( A \) is given by:

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

Substituting the values we calculated:

\[ P(\text{only A}) = \frac{15}{86} \]

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

\[ \frac{15}{86} \]

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