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To count: The number of four-digit numbers whose digits are 1, 2, 3, 4, or 5 and no property holds.
Answer to Problem 1E
The number of four digit numbers is 625.
Explanation of Solution
Procedure used:
Multiplication principle:
When a task has p outcomes and, no matter what the outcome of the first task, a second task has some q outcomes, then the two tasks performed consecutively will have
Calculation:
It is given that the four-digit number has to be formed from amongst the five digits.
As no property holds, each digit place of the four-digit number will be any of the five digits.
Hence, the number of four digit numbers becomes
Therefore, the number of four digit numbers is 625.
To count: The number of four-digit numbers whose digits are 1, 2, 3, 4, or 5 and the property of the digits being distinct, holds.
Answer to Problem 1E
The number of four digit numbers is 120.
Explanation of Solution
Procedure used:
Multiplication principle:
When a task has p outcomes and, no matter what the outcome of the first task, a second task has some q outcomes, then the two tasks performed consecutively will have
Calculation:
It is given that the four-digit number has to be formed from amongst the five digits.
As property of the digit being distinct holds, each digit place of the four-digit number will be any of the five digits but one less from the previous in the next place.
Hence, the number of four digit numbers becomes
Therefore, the number of four digit numbers is 120.
To count: The number of four-digit numbers whose digits are 1, 2, 3, 4, or 5 and the property of the number being even.
Answer to Problem 1E
The number of four digit numbers is 250.
Explanation of Solution
Procedure used:
Multiplication principle:
When a task has p outcomes and, no matter what the outcome of the first task, a second task has some q outcomes, then the two tasks performed consecutively will have
Calculation:
It is given that the four-digit number has to be formed from amongst the five digits.
As the property of the number being even holds, each digit place of the four-digit number will be any of the five digits except at unit’s place.
The unit’s place of the numbers will carry even digits that are 2 and 4.
Hence, the number of four digit numbers becomes
Therefore, the number of four digit numbers is 250.
To count: The number of four-digit numbers whose digits are 1, 2, 3, 4, or 5 and both the properties hold.
Answer to Problem 1E
The number of four digit numbers is 48.
Explanation of Solution
Procedure used:
Multiplication principle:
When a task has p outcomes and, no matter what the outcome of the first task, a second task has some q outcomes, then the two tasks performed consecutively will have
Calculation:
It is given that the four-digit number has to be formed from amongst the five digits.
As both the properties hold, each digit place of the four-digit number will be any of the five digits.
However, the choices for the unit’s place are 2 and the choices for other places starts from 4 and ends at 2.
Hence, the number of four digit numbers becomes
Therefore, the number of four digit numbers is 48.
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Chapter 2 Solutions
Introductory Combinatorics
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