From `10` identical cards, numbered `1,2,3,…., 10` one card is drawn at random. Find the probability that the number on the card drawn is a multiple of:
`2` and `3`

Options

A. `(1)/(10)`
B. `(3)/(10)`
C. `(1)/(11)`
D. `(1)/(7)`

Answer

To solve the problem of finding the probability that the number on the card drawn is a multiple of both `2` and `3`, we can follow these steps:

### Step 1: Identify the Total Number of Outcomes
The total number of outcomes is the total number of cards available. Since the cards are numbered from `1` to `10`, the total number of outcomes is:

\[ \text{Total Outcomes} = 10 \]

### Step 2: Identify the Multiples of 2 and 3
Next, we need to find the numbers between `1` and `10` that are multiples of both `2` and `3`. The least common multiple (LCM) of `2` and `3` is `6`.

Now, we will check the multiples of `6` within the range of `1` to `10`:

- The multiples of `6` are: `6`

### Step 3: Count the Favorable Outcomes
From the previous step, we see that there is only one number (`6`) that is a multiple of both `2` and `3`. Therefore, the number of favorable outcomes is:

\[ \text{Favorable Outcomes} = 1 \]

### Step 4: Calculate the Probability
The probability \( P \) of drawing a card that is a multiple of both `2` and `3` is given by the formula:

\[ P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{1}{10} \]

### Final Answer
Thus, the probability that the number on the card drawn is a multiple of `2` and `3` is:

\[ \boxed{\frac{1}{10}} \]

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