How Many Different 4-Digit Numbers Can Be Formed Using Digits 1, 3, 3, 7, 7, and 8?
To determine how many different 4-digit numbers can be formed with the digits 1, 3, 3, 7, 7, and 8, we need to carefully analyze the possible arrangements of these digits, taking into account the repetitions.
Breaking Down the Problem
The solution involves breaking down the problem into various cases based on the number of repeated digits and calculating the permutations for each case.
Case 1: All Digits are Different
The only set of digits that fit this category is {1, 3, 7, 8}. Since all four digits are unique, the number of ways to arrange these digits is given by the factorial function:
4! 4 × 3 × 2 × 1 24
Case 2: One Digit Appears Twice and Two Other Digits are Different
We can have two scenarios in this case: either the digit 3 appears twice or the digit 7 appears twice. We will analyze each sub-case separately.
Sub-Case 2a: The Digit 3 Appears Twice
The digits used will be {3, 3, 1, 7} or {3, 3, 1, 8} or {3, 3, 7, 8}. The number of arrangements for {3, 3, A, B} , where A and B are different, is given by:
frac{4!}{2!} frac{24}{2} 12
Since there are 3 different combinations, the total for this sub-case is:
12 × 3 36
Sub-Case 2b: The Digit 7 Appears Twice
The digits used will be {7, 7, 1, 3} or {7, 7, 1, 8} or {7, 7, 3, 8}. Similarly, the number of arrangements for {7, 7, A, B} is:
frac{4!}{2!} frac{24}{2} 12
Again, since there are 3 different combinations, the total for this sub-case is:
12 × 3 36
Case 3: Two Digits Appear Twice
The only possibility here is {3, 3, 7, 7}. The number of arrangements for this combination is:
frac{4!}{2! × 2!} frac{24}{4} 6
Total Count
Adding all the cases together, we get:
24 36 36 6 102
Therefore, the total number of different 4-digit numbers that can be formed is 102.
Alternative Analysis
An alternative way to approach this problem is to start with the total number of combinations with six digits and then subtract the combinations that do not form valid 4-digit numbers. Here’s the step-by-step breakdown:
Total combinations with six digits: 6 × 5 × 4 × 3 360 Number of 4-digit combinations with four unique digits: 1378 24 So, there are 360 - 24 336 combinations with duplicate digits. Now, we need to calculate how many of these are duplicate numbers that we need to deduct: Case 1a: 24 combinations but only 12 unique (deduct 12). Case 1b: 24 combinations but only 12 unique (deduct 12). Case 2: 24 combinations all unique (no deduction needed). Case 3: 120 combinations but only 24 unique (deduct 96). Case 4: Same as Case 3 (deduct 96). Case 5: 24 combinations but only 6 unique (deduct 18).In total, we need to deduct 12 12 96 96 18 234 from 336, leaving us with 336 - 234 102 .
Thus, the total number of different 4-digit numbers that can be formed is 102, as confirmed by both methods.