Forming 5-Digit Numbers Using Digits 3, 5, 7, 9

The problem asks us to find the number of 5-digit numbers that can be formed using the digits 3, 5, 7, and 9, with the condition that each of these four digits must appear at least once in the 5-digit number.

We are forming a 5-digit number using a set of 4 distinct digits {3, 5, 7, 9}. Since the number has 5 digits and we must use all 4 distinct digits, exactly one of the digits must be repeated.

Let the five positions for the digits be P1, P2, P3, P4, P5. We need to fill these positions using the digits 3, 5, 7, 9 such that all four digits are present. This means the set of 5 digits used will contain the four distinct digits {3, 5, 7, 9} plus one extra digit, which must be a repeat of one of the four.

There are 4 possibilities for which digit is repeated:

• Digit 3 is repeated. The digits used are {3, 3, 5, 7, 9}.
• Digit 5 is repeated. The digits used are {3, 5, 5, 7, 9}.
• Digit 7 is repeated. The digits used are {3, 5, 7, 7, 9}.
• Digit 9 is repeated. The digits used are {3, 5, 7, 9, 9}.

Now, for each case, we need to find the number of distinct permutations of the 5 digits. The formula for permutations of $n$ objects where one object appears $r$ times is $\frac{n!}{r!}$. In our cases, $n=5$ (total digits) and $r=2$ (the repeated digit appears twice).

Let's calculate the number of arrangements for each case:

Since these four cases are mutually exclusive (a number cannot have both 3 and 5 as the only repeated digit), the total number of 5-digit numbers formed by these digits where each of the four digits appears is the sum of the numbers of permutations in each case.

Total number of 5-digit numbers = (Permutations when 3 is repeated) + (Permutations when 5 is repeated) + (Permutations when 7 is repeated) + (Permutations when 9 is repeated)

Total number = $60 + 60 + 60 + 60 = 4 \times 60 = 240$.

Thus, there are 240 such 5-digit numbers.

Revision Table: Key Concepts

Additional Information: Combinations vs. Permutations

It's important to understand the difference between combinations and permutations when solving problems like this.

• Combinations: Refer to the selection of objects where the order does not matter. For example, choosing a committee of 3 people from a group of 10.
• Permutations: Refer to the arrangements of objects where the order does matter. For example, arranging 3 people in a line or forming a number using digits.

In this problem, we are forming 5-digit numbers, and the order of the digits matters (e.g., 33579 is different from 97533). Therefore, we use permutations.

The condition that each of the four digits (3, 5, 7, 9) must appear means we cannot form numbers using only three or fewer distinct digits from the given set. Since we need a 5-digit number, and we start with 4 distinct digits, one digit must necessarily be used twice. Identifying this repeated digit is the first step in setting up the cases for permutations.