Counting Numbers between 3000 and 4000 using Permutations

To count the number of numbers that can be formed between 3000 and 4000 using the digits 1,2,3,4,5, and 6, we have to use permutations. Permutations are arrangements of objects where order matters.

Step 1: Find the number of digits we can choose for the thousands place.

Since the numbers have to be between 3000 and 4000, the first digit can only be 3. Therefore, we do not need to consider the permutations of 1, 2, 4, 5, and 6 for the thousands place. So, we have only one choice for the thousands place, which is 3.

Step 2: Find the number of digits we can choose for the hundreds place.

We have six digits to choose from for the hundreds place since we have not used any digit for this place yet. Therefore, we have 6 choices for the hundreds place.

Step 3: Find the number of digits we can choose for the tens place.

We still have five digits to choose from for the tens place since we have used two digits for the thousands and hundreds places. Therefore, we have 5 choices for the tens place.

Step 4: Find the number of digits we can choose for the ones place.

We still have four digits to choose from for the ones place since we have used three digits for the thousands, hundreds, and tens places. Therefore, we have 4 choices for the ones place.

Step 5: Multiply the number of choices for each place.

To find the total number of numbers between 3000 and 4000 that can be formed using the digits 1,2,3,4,5, and 6, we have to multiply the number of choices for each place.

Total number of numbers = 1 x 6 x 5 x 4 = 120

Therefore, there are 120 numbers between 3000 and 4000 that can be formed using the digits 1,2,3,4,5, and 6.