To determine how many different three-digit numbers can be formed using the digits from the set {2, 3, 4, 5, 6} without any repeating digits, we use the concept of permutations. Here's a step-by-step approach to solve this problem:
1. Identify the total number of digits available:
The set provided is {2, 3, 4, 5, 6}, which contains 5 distinct digits.
2. Recognize the requirement for the three-digit number:
We need to form numbers using exactly 3 of these 5 digits.
3. Apply the permutation formula:
When forming a sequence where order matters and no repeats are allowed, it's a permutation. The number of permutations of 5 digits taken 3 at a time is given by the formula:
[tex]\[
P(n, r) = \frac{n!}{(n - r)!}
\][/tex]
where [tex]\( n \)[/tex] is the total number of items to choose from and [tex]\( r \)[/tex] is the number of items to choose.
4. Substitute the values into the formula:
Here, [tex]\( n = 5 \)[/tex] (total available digits) and [tex]\( r = 3 \)[/tex] (digits to choose).
[tex]\[
P(5, 3) = \frac{5!}{(5 - 3)!} = \frac{5!}{2!}
\][/tex]
5. Calculate the factorial values:
- [tex]\( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)[/tex]
- [tex]\( 2! = 2 \times 1 = 2 \)[/tex]
6. Divide the factorial values to find the number of permutations:
[tex]\[
P(5, 3) = \frac{120}{2} = 60
\][/tex]
Therefore, the number of different three-digit numbers that can be written using the digits from the set {2, 3, 4, 5, 6} without any repeating digits is [tex]\( \boxed{60} \)[/tex].
