Answer to Question #325620 in Discrete Mathematics for shadi

Question

Consider the design of a communication system.

⦁ How many three-digit phone prefixes that are used to represent a particular geographic area (such as an area code) can be created from the digits 0 through 9?

⦁ As in part i., how many three-digit phone prefixes are possible that do not start with 0 or 1, but contain 0 or 1 as the middle digit?

⦁ How many three-digit phone prefixes are possible in which no digit appears more than once in each prefix?

Answer 1

a. From the multiplication rule (or by realizing that this is ordered sampling with 

replacement), 10³= 1,000 prefixes are 

possible.

b.

From the multiplication rule, 8 · 2 · 10 =160 prefixes are possible.

8 - because it is digits 2,3,4,5,6,7,8,9 -this is start number

2 -because it is 0 or 1 - middle number

10 - because it is any digit from 0,1,2,3,4,5,6,7,8,9

c.

This is ordered sampling without replacement.

Therefore $P^3_{10}=\frac{10!}{(10-3)!}=10x9x8=720$

prefixes are possible.