To find the number of 3-digit numbers that can be formed using numerals in the set [tex]\(\{3, 2, 7, 9\}\)[/tex] without repetition, we can follow these steps:
### Step 1: Identify the parameters
- The numerals given are [tex]\(\{3, 2, 7, 9\}\)[/tex].
- We need to form 3-digit numbers.
- Repetition of numerals is not allowed.
### Step 2: Define [tex]\( n \)[/tex] and [tex]\( r \)[/tex]
- [tex]\( n \)[/tex] is the number of available numerals, which is 4.
- [tex]\( r \)[/tex] is the number of digits we want to choose to form a 3-digit number, which is 3.
### Step 3: Use the formula for permutations without repetition
The formula for permutations without repetition is:
[tex]\[
{}_nP_r = \frac{n!}{(n-r)!}
\][/tex]
### Step 4: Calculate the factorials
Using [tex]\( n = 4 \)[/tex] and [tex]\( r = 3 \)[/tex]:
1. Calculate [tex]\( n! \)[/tex]:
[tex]\[
4! = 4 \times 3 \times 2 \times 1 = 24
\][/tex]
2. Calculate [tex]\( (n-r)! \)[/tex]:
[tex]\[
(4-3)! = 1! = 1
\][/tex]
### Step 5: Apply the formula
[tex]\[
{}_4P_3 = \frac{4!}{(4-3)!} = \frac{24}{1} = 24
\][/tex]
### Step 6: Conclusion
The number of 3-digit numbers that can be formed using the numerals [tex]\(\{3, 2, 7, 9\}\)[/tex] without repetition is:
[tex]\[
24
\][/tex]
Thus, 24 different 3-digit numbers can be formed.
