Cards are numbered 1 to 25 . Two cards are drawn at random . Find the probability that number on one card is a multiple of 7 and on the other is a multiple of 11.

Answer

To solve the problem of finding the probability that one card drawn is a multiple of 7 and the other is a multiple of 11, we can follow these steps:

### Step 1: Identify the total number of cards
The total number of cards is given as 25.

### Step 2: Identify the multiples of 7
The multiples of 7 between 1 and 25 are:
- 7 (7 × 1)
- 14 (7 × 2)
- 21 (7 × 3)

Thus, there are **3 multiples of 7**.

### Step 3: Identify the multiples of 11
The multiples of 11 between 1 and 25 are:
- 11 (11 × 1)
- 22 (11 × 2)

Thus, there are **2 multiples of 11**.

### Step 4: Calculate the total ways to choose 2 cards from 25
The total number of ways to choose 2 cards from 25 is given by the combination formula:
\[ \text{Total ways} = \binom{25}{2} = \frac{25 \times 24}{2} = 300 \]

### Step 5: Calculate the favorable outcomes
We need to consider two scenarios:
1. The first card is a multiple of 7 and the second card is a multiple of 11.
2. The first card is a multiple of 11 and the second card is a multiple of 7.

#### Scenario 1: First card is a multiple of 7, second is a multiple of 11
- The number of ways to choose the first card (multiple of 7) = 3
- The number of ways to choose the second card (multiple of 11) = 2

Thus, the total ways for this scenario = \(3 \times 2 = 6\).

#### Scenario 2: First card is a multiple of 11, second is a multiple of 7
- The number of ways to choose the first card (multiple of 11) = 2
- The number of ways to choose the second card (multiple of 7) = 3

Thus, the total ways for this scenario = \(2 \times 3 = 6\).

### Step 6: Calculate the total favorable outcomes
The total favorable outcomes = Outcomes from Scenario 1 + Outcomes from Scenario 2 = \(6 + 6 = 12\).

### Step 7: Calculate the probability
The probability \(P\) that one card is a multiple of 7 and the other is a multiple of 11 is given by:
\[ P = \frac{\text{Total favorable outcomes}}{\text{Total ways to choose 2 cards}} = \frac{12}{300} = \frac{1}{25} \]

### Final Answer
The probability that one card is a multiple of 7 and the other is a multiple of 11 is \(\frac{1}{25}\).

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