Given the set of numbers {1,2,3,4,5,6,7}, a. How many...

Given the set of numbers $\{1,2,3,4,5,6,7\}$, a. How many different four-digit numbers can be formed? b. How many different four-digit numbers can be formed if the number cannot have repeated digits? c. How many different four-digit numbers can be formed if the number is to be divisible by 5 and repetition of digits is allowed?

Given the set of numbers $\{1,2,3,4,5,6,7\}$,
a. How many different four-digit numbers can be formed?
b. How many different four-digit numbers can be formed if the number cannot have repeated digits?
c. How many different four-digit numbers can be formed if the number is to be divisible by 5 and repetition of digits is allowed?

Key Concepts

Multiplication Principle

The multiplication principle is a foundational concept in combinatorics stating that if one event can occur in m ways and another independent event can occur in n ways, then the two events together can occur in m times n ways. This principle is used to count the total number of outcomes when forming sequences such as four-digit numbers, by multiplying the number of choices available at each step.

Permutations Without Repetition

Permutations without repetition involve arranging items in a specific order such that no item is used more than once. In problems where repetition is not allowed, after choosing a digit for one position, it cannot be reused in another position, reducing the number of available choices for subsequent positions. This concept is critical when calculating the number of four-digit numbers formed from a set where each digit must be unique.

Counting With Repetition Allowed

Counting with repetition allowed means that each digit in a sequence can be chosen independently from the set, even if it has been used before. This allows for every position in the four-digit number to have the full set of choices available, simplifying the calculation by using the multiplication principle for each digit position.

Applying Divisibility Constraints

Divisibility constraints in combinatorial problems require that the numbers formed adhere to specific mathematical properties, such as being divisible by 5. This typically influences the selection of one or more digit positions—for example, for a number to be divisible by 5, the last digit must satisfy a particular condition. Incorporating such constraints means reducing the possibilities for the affected positions while applying the counting principles to the remainder of the positions.

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