How many four-digit numbers can be formed using the elements of the set one, two, three, seven, and nine?
One way of answering this question is using the fundamental counting principle. This states that if there are P ways to do one thing and Q ways to do a second thing, there are P multiplied by Q ways to do both things. In this question, we are trying to create a four-digit number. There are five elements in the set, and there is no restriction on repetition. We can choose any one of the five elements one, two, three, seven, and nine for the first digit. We can also choose any one of the five elements for the second digit. Likewise, there are five possible choices for the third digit, and the same is true for the fourth digit.
The total number of four-digit numbers from the set will therefore be equal to five multiplied by five multiplied by five multiplied by five. Five multiplied by five is equal to 25. And multiplying 25 by 25 gives us 625. There are 625 four-digit numbers that could be formed from the elements of the set one, two, three, seven, and nine.
An alternative method here would be to use our knowledge of permutations. When calculating the total number of permutations with replacement, we use the formula 𝑛 to the power of 𝑟. In this case, the value of 𝑛 is the number of elements in the set, and the value of 𝑟 is the number of digits. We need to calculate five to the fourth power or five to the power of four. Once again, this gives us an answer of 625.