To determine how many different four-digit numbers can be made using the digits [tex]\(1, 2, 3, 4, 5, 6\)[/tex] without repeating any digits, follow these steps:
1. Identify the Total Number of Digits (n):
The total number of available digits is 6 (i.e., [tex]\(1, 2, 3, 4, 5, 6\)[/tex]).
2. Identify the Number of Digits in Each Number (r):
We need to form four-digit numbers, so [tex]\(r = 4\)[/tex].
3. Understand the Concept of Permutations:
Because the order of digits matters and no digit can be used more than once, we need to use permutations rather than combinations. The number of permutations of [tex]\(n\)[/tex] items taken [tex]\(r\)[/tex] at a time is denoted as [tex]\({}_nP_r\)[/tex].
4. Apply the Permutation Formula:
The formula for permutations is given by:
[tex]\[
{}_nP_r = \frac{n!}{(n-r)!}
\][/tex]
where [tex]\(n!\)[/tex] is the factorial of [tex]\(n\)[/tex], and [tex]\((n-r)!\)[/tex] is the factorial of [tex]\((n-r)\)[/tex].
5. Substitute the Values of [tex]\(n\)[/tex] and [tex]\(r\)[/tex]:
Here, [tex]\(n = 6\)[/tex] and [tex]\(r = 4\)[/tex]. Substitute these values into the permutation formula:
[tex]\[
{}_6P_4 = \frac{6!}{(6-4)!} = \frac{6!}{2!}
\][/tex]
6. Calculate the Factorials:
- [tex]\(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)[/tex]
- [tex]\(2! = 2 \times 1 = 2\)[/tex]
7. Compute the Permutation:
[tex]\[
{}_6P_4 = \frac{720}{2} = 360
\][/tex]
Therefore, the number of different four-digit numbers that can be made using the digits [tex]\(1, 2, 3, 4, 5, 6\)[/tex] without repeating any digit is 360.
The correct answer is:
c. [tex]\({}_6P_4 = 360\)[/tex]
