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Chapter 2: Problem 74

How many different numbers of 3 digits can be formed from the numbers \(1,2,3,4,5\) (a) If repetitions are allowed? (b) If repetitions are not allowed? How many of these numbers are even in either case?

Short Answer

Expert verified
With repetitions allowed, there are 125 different 3-digit numbers, of which 50 are even. Without repetitions, there are 60 different 3-digit numbers, with 24 being even.

Step by step solution

01

Identify the total number of 3-digit numbers

Since repetitions are allowed, we can use each of the 5 digits for any of the 3 positions in the number. Thus, for each position, we have 5 possibilities.
02

Calculate the total number of 3-digit numbers with repetitions allowed

We have \(5 \times 5 \times 5\) possible combinations which, when calculated, give us a total of \(5^3 = 125\) different 3-digit numbers.
03

Determine the total number of even numbers

For a number to be even, it must end with an even digit, which in this case are 2 and 4. Since there are 2 even digits and 5 choices for each of the other two positions, we have \(5 \times 5 \times 2\) even numbers, which give us a total of \(5^2 \times 2 = 50\) even numbers. Case (b): Repetitions are not allowed
04

Identify the total number of 3-digit numbers without repetitions

Since repetitions are not allowed, we have less choice for each subsequent position. For the first position, we can still choose from 5 digits. For the second position, we have 4 choices left, and for the third position, 3 choices remain.
05

Calculate the total number of 3-digit numbers without repetitions

We have \(5 \times 4 \times 3\) possible combinations, which, when calculated, give us a total of \(5!/(5-3)! = 60\) different 3-digit numbers.
06

Determine the total number of even numbers without repetitions

For a number to be even without repetitions, we still need it to end with 2 or 4. So, there are 2 choices for the even digit, 4 choices for the first position (since we've used an even digit at the end already), and 3 choices for the second position. Therefore, we have \(4 \times 3 \times 2 = 24\) even numbers. So, in summary: (a) Repetitions allowed: - Total different 3-digit numbers: 125 - Even numbers: 50 (b) Repetitions not allowed: - Total different 3-digit numbers: 60 - Even numbers: 24

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