From 25 tickets , marked with the first 25 numberals , one is drawn at random . Find the probalility that
it is a multiple of 3 or 7.

Answer

To solve the problem, we need to find the probability of drawing a ticket that is a multiple of either 3 or 7 from a total of 25 tickets numbered from 1 to 25.

### Step-by-Step Solution:

1. **Identify the Total Outcomes:**
The total number of tickets is 25 (numbered from 1 to 25).

**Hint:** The total outcomes are simply the total number of tickets available.

2. **Find the Multiples of 3:**
The multiples of 3 within the range of 1 to 25 are:
- 3, 6, 9, 12, 15, 18, 21, 24

Count of multiples of 3 = 8

**Hint:** To find multiples of a number, divide the maximum number by that number and count the integers.

3. **Find the Multiples of 7:**
The multiples of 7 within the range of 1 to 25 are:
- 7, 14, 21

Count of multiples of 7 = 3

**Hint:** Similar to multiples of 3, divide the maximum number by 7 and list the integers.

4. **Identify Common Multiples (Multiples of Both 3 and 7):**
The common multiples of 3 and 7 (i.e., multiples of 21) within the range of 1 to 25 are:
- 21

Count of common multiples = 1

**Hint:** Use the Least Common Multiple (LCM) to find common multiples.

5. **Use the Inclusion-Exclusion Principle:**
To find the total number of favorable outcomes (multiples of 3 or 7), we use the formula:
\[ \text{Total favorable outcomes} = (\text{Multiples of 3}) + (\text{Multiples of 7}) - (\text{Common multiples}) \]
Substituting the values:
\[ \text{Total favorable outcomes} = 8 + 3 - 1 = 10 \]

**Hint:** This principle helps avoid double counting the common multiples.

6. **Calculate the Probability:**
The probability \( P \) of drawing a ticket that is a multiple of 3 or 7 is given by:
\[ P = \frac{\text{Total favorable outcomes}}{\text{Total outcomes}} = \frac{10}{25} \]
Simplifying this fraction gives:
\[ P = \frac{2}{5} \]

**Hint:** Probability is always expressed as a fraction of favorable outcomes over total outcomes.

### Final Answer:
The probability that the drawn ticket is a multiple of 3 or 7 is \( \frac{2}{5} \).