The digits, from `0 to 9` are written on `10` slips of paper (one digit on each slip) and placed in a box. If three of the slips are drawn and arranged, then the number of possible different arrangements is

Options

A. `1000`
B. `720`
C. `810`
D. None of these

Explanation

To solve the problem of finding the number of different arrangements of three slips drawn from a set of ten slips (numbered from 0 to 9), we can use the concept of permutations. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the problem
We need to draw 3 slips from a total of 10 slips, each labeled with a unique digit from 0 to 9. After drawing the slips, we will arrange them.

### Step 2: Identify the formula for permutations
The number of ways to arrange \( r \) items selected from \( n \) items is given by the formula for permutations:
\[ nP_r = \frac{n!}{(n - r)!} \]
where \( n! \) (n factorial) is the product of all positive integers up to \( n \).

### Step 3: Assign values to \( n \) and \( r \)
In this problem:
- \( n = 10 \) (the total number of slips)
- \( r = 3 \) (the number of slips we want to arrange)

### Step 4: Substitute into the permutation formula
Now we substitute \( n \) and \( r \) into the permutations formula:
\[ 10P3 = \frac{10!}{(10 - 3)!} = \frac{10!}{7!} \]

### Step 5: Simplify the expression
We can simplify \( \frac{10!}{7!} \):
\[ 10! = 10 \times 9 \times 8 \times 7! \]
Thus,
\[ 10P3 = \frac{10 \times 9 \times 8 \times 7!}{7!} = 10 \times 9 \times 8 \]

### Step 6: Calculate the result
Now we calculate \( 10 \times 9 \times 8 \):
\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]

### Final Answer
The number of possible different arrangements of the three slips drawn is \( 720 \).

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