The number of 4 digit numbers formed with the digits 1, 1, 2, 2, 3, 4 isa)100b)101c)201d)none of theseCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

Answer

Given: Digits 1, 1, 2, 2, 3, 4

To find: Number of 4 digit numbers that can be formed using these digits

Solution:

We can form 4 digit numbers in the following ways:

1. All digits are different

In this case, we can choose the thousands place in 4 ways (since 0 cannot be used), the hundreds place in 3 ways (since we have used 1 of the digits already), the tens place in 2 ways (since we have used 2 of the digits already) and the units place in 1 way (since we have used 3 of the digits already). Therefore, the number of such 4 digit numbers is 4 x 3 x 2 x 1 = 24.

2. Two digits are the same, the other two are different

In this case, we can choose the repeated digit in 3 ways (since we have 3 digits that are repeated), the non-repeated digit in 3 ways (since we have 3 digits that are not repeated), the thousands place in 2 ways (since we have used 1 of the digits already), and the remaining place in 2 ways (since we have used 2 of the digits already). Therefore, the number of such 4 digit numbers is 3 x 3 x 2 x 2 = 36.

Total number of 4 digit numbers = 24 + 36 = 60

Therefore, the correct answer is option D, none of these, as 60 is not one of the given options.

Answer

Digits : 1,1,2,2,3,4
so, total digits is 6
we have to form a 4 digit number: 4
= ⁶P4
=6×5×4×3
=360

same digits: 1,1,2,2
so, 2! 2! = 4

360/4 = 90

Option (D) none of these is correct

Answer

Correct ans is 9

Answer

We are asked to find the number of 4-digit numbers that can be formed using the digits 1, 1, 2, 2, 3, 4.

Step 1: Understanding the Problem
We need to form a 4-digit number using the digits 1, 1, 2, 2, 3, 4, and we can repeat digits as long as they are available. The key thing to note is that some of the digits are repeated (1 and 2 each appear twice), so we need to consider this repetition when calculating the total number of different 4-digit numbers.

Step 2: Case Analysis
We can form the 4-digit number by choosing different combinations of digits. Since there are 6 digits in total but only 4 digits need to be selected, we can have multiple cases based on the selection of digits:

Case 1: Two 1’s and Two 2’s
If we choose two 1's and two 2's, we need to arrange the four digits: 1, 1, 2, 2. The number of distinct arrangements of these digits is calculated as:

Case 2: Two 1’s and One 2, One 3
If we choose two 1's, one 2, and one 3, we need to arrange the digits: 1, 1, 2, 3. The number of distinct arrangements of these digits is:

Case 3: Two 1’s and One 2, One 4
If we choose two 1's, one 2, and one 4, we need to arrange the digits: 1, 1, 2, 4. The number of distinct arrangements of these digits is:

Case 4: One 1, One 2, One 3, One 4
If we choose one 1, one 2, one 3, and one 4, we need to arrange the digits: 1, 2, 3, 4. The number of distinct arrangements of these digits is:
4! = 24
Step 3: Total Number of Arrangements
Now, we add up all the distinct arrangements from each case:
6+12+12+24=54
The total number of distinct 4-digit numbers that can be formed is 54.

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