Since we are trying to find the number of sequences can be made without repetition, we are going to use a combination.
The formula for combinations is:
[tex]_n C _k = \dfrac{n!}{k! (n - k)!}[/tex]
- [tex]n[/tex] is the total number of elements in the set
- [tex]k[/tex] is the number of those elements you are desiring
Since there are 10 total digits, [tex]n = 10[/tex] in this scenario. Since we are choosing 6 digits of the 10 for our sequence, [tex]k = 6[/tex] in this scenario. Thus, we are trying to find [tex]_{10} C _6[/tex]. This can be found as shown:
[tex]_{10} C _6 = \dfrac{10!}{6! \cdot 4!} = \dfrac{10 \cdot 9 \cdot 8 \cdot 7}{4!} = \dfrac{5040}{24} = 210[/tex]
There are 210 total combinations.
