Find the probability that number selected from 1 to 50 such that number is divisible by at least by 4, 6 or 7

Options

A. `21/50`
B. `1/2`
C. `19/50`
D. `23/50`

Answer

To find the probability that a number selected from 1 to 50 is divisible by at least one of the numbers 4, 6, or 7, we can use the principle of inclusion-exclusion.

### Step-by-Step Solution:

1. **Identify the Total Numbers**:
The total number of integers from 1 to 50 is 50.

2. **Count Numbers Divisible by Each Number**:
- **Divisible by 4**:
The numbers divisible by 4 from 1 to 50 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48.
There are 12 such numbers.
- **Divisible by 6**:
The numbers divisible by 6 from 1 to 50 are: 6, 12, 18, 24, 30, 36, 42, 48.
There are 8 such numbers.
- **Divisible by 7**:
The numbers divisible by 7 from 1 to 50 are: 7, 14, 21, 28, 35, 42, 49.
There are 7 such numbers.

3. **Count Numbers Divisible by the Pairwise Intersections**:
- **Divisible by both 4 and 6 (LCM = 12)**:
The numbers are: 12, 24, 36, 48.
There are 4 such numbers.
- **Divisible by both 4 and 7 (LCM = 28)**:
The numbers are: 28.
There is 1 such number.
- **Divisible by both 6 and 7 (LCM = 42)**:
The numbers are: 42.
There is 1 such number.

4. **Count Numbers Divisible by All Three (4, 6, and 7)**:
- **Divisible by 4, 6, and 7 (LCM = 84)**:
There are no numbers between 1 and 50 that are divisible by 84, so this count is 0.

5. **Apply Inclusion-Exclusion Principle**:
Using the formula:
\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \]
where:
- \( |A| = 12 \) (divisible by 4)
- \( |B| = 8 \) (divisible by 6)
- \( |C| = 7 \) (divisible by 7)
- \( |A \cap B| = 4 \) (divisible by both 4 and 6)
- \( |A \cap C| = 1 \) (divisible by both 4 and 7)
- \( |B \cap C| = 1 \) (divisible by both 6 and 7)
- \( |A \cap B \cap C| = 0 \) (divisible by 4, 6, and 7)

Plugging in the values:
\[ |A \cup B \cup C| = 12 + 8 + 7 - 4 - 1 - 1 + 0 = 21 \]

6. **Calculate the Probability**:
The probability \( P \) that a number selected from 1 to 50 is divisible by at least one of 4, 6, or 7 is given by:
\[ P = \frac{|A \cup B \cup C|}{\text{Total Numbers}} = \frac{21}{50} \]

### Final Answer:
The probability that a number selected from 1 to 50 is divisible by at least one of 4, 6, or 7 is \( \frac{21}{50} \).