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Chapter 33: Problem 1066

How many telephone numbers of four different digits each can be made from the digits \(0,1,2,3,4,5,6,7,8,9 ?\)

Short Answer

Expert verified
There are 4536 telephone numbers of four different digits each that can be made from the given digits, considering that the numbers must be distinct and the first digit cannot be 0.

Step by step solution

01

Understanding Permutations

Permutations refer to the arrangement of objects in a specific order. In this case, we are arranging 4 digits out of the given 10 digits. Since we are choosing distinct digits, we can use the formula for permutations without repetition.
02

Apply the permutation formula

The formula for permutations without repetition is: \(P(n, r) = \frac{n!}{(n-r)!}\) Where: - n is the number of elements to choose from - r is the number of elements chosen - ! denotes the factorial (e.g., 5! = 5 脳 4 脳 3 脳 2 脳 1 = 120) Since we have 9 options for the first digit (1-9) and 9 remaining options for the second digit (0 and the remaining 8 digits not chosen in the first step), we need to calculate permutations in two stages: 1. Choosing the first digit (9 options) 2. Choosing the second, third, and fourth digits (9 options each)
03

Calculate the permutations for each stage

1. Permutations for choosing the first digit (from 1-9): \(P(9, 1) = \frac{9!}{(9-1)!} = \frac{9!}{8!} = 9\) 2. Permutations for choosing the second, third, and fourth digits (9 options): Since we have 9 options for each position, we can first calculate the permutation for three digits and then multiply by 9. \(P(9, 3) = \frac{9!}{(9-3)!} = \frac{9!}{6!} = 9 脳 8 脳 7\)
04

Multiply the permutations

Now, multiply the permutations from Step 3 to get the total number of unique 4-digit telephone numbers: Total permutations = Permutations for first digit 脳 Permutations for remaining digits Total permutations = 9 脳 9 脳 8 脳 7 = 4536 Hence, there are 4536 telephone numbers of four different digits each that can be made from the given digits.

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