Combinatorics - Finding all the four digit numbers with a property

In summary, the conversation is about a textbook question on permutation and the steps taken to solve it. The question asks for the number of four digit numbers that can be constructed using the digits {1,2,3,4,5,6,7,8,9} without any digit repetition and above 3400. The answer provided is 2340, calculated by considering numbers between 3000 and 4000 and numbers above 4000 separately. The person asking for verification of their method later confirms that they have arrived at the correct answer.
  • #1
Sleek
60
0
Hello,

Recently, I've come across a textbook question based on permutation. I've even got the answer to that, but i want to verify my steps, because each time i solved the question, i realized how tricky it was, and i had to modify my approach. Heres my final approach, please spot any mistakes you find.

Question : There are 9 digits, namely {1,2,3,4,5,6,7,8,9}. The thing that has to be found out is, the number of four digit numbers which can be constructed from those digits such that no digits are repeated in the number and all the numbers are above/greater than 3400. This sum is quite tricky, as if it would have been a number like 4000, then it might be easy to use the Fundamental Principle.

Answer: : Heres my attempted answer.

By using the digits from 1 to 9 and between 3000 and 4000, the least number above 3400, without any repitition of digits which can be constructed is 3412. The greatest is 3987.

From the above two numbers, we can see that,

-> The first place (from the left) only assumes the digit 3. So the number of digits it can be substituted with is 1.
-> The second place, can assume 4,5,6,7,8,9. So the number of digits it can be substituted with is 6.
-> The third place assumes digits from 1 to 8. So no. of combinitions is 8.
-> The fourth place assumes digits 2,3,4,5,6,7. So no. of combinitions is 6.

By using fundamental principle, all four digits numbers between 3000 and 4000 such that, they only have digits 1 to 9, no digit is repeated, is 1*6*8*6 = 288.

Now let's consider all digits above 4000. The reason this is done seperately is because, between 3000 and 4000, the second or/and other places can assume less no. of digits as opposed to cases above 4000.

For all no. above 4000,

-> The first place (from left) can only assume digits 4,5,6,7,8,9. So no. of combinitions is 6.
-> The second, third and fourth place can assume digits 8,7,6 as repitition of digits is not allowed.

So by fundamental principle, no of possible digits above 4000, with no digits repeated is 6*8*7*6 = 2016.

If we add up 288 and 2016, we get 2340, which seems to be the required answer for all possible permutations of 4 digit number using the digits 1 to 9, with no repitition of digits allowed in the number, and above 3400.

The Above 4000 Part was a sum by itself, and the answer to that part is right. But i am concerned about my approach of seperating the sections in 3000 to 4000 and above 4000. Can anyone verify my method?


Regards,
Sleek.
 
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  • #2
Hi again,

I've managed to get the right answer. But thanks anyways :smile: .

Regards,
Sleek.
 
  • #3


Dear Sleek,

Your approach seems to be correct. By separating the numbers above and below 4000, you have taken into account the different restrictions and combinations of digits that can be used in each case. Your final answer of 2340 does seem to be the correct number of possible permutations based on the given criteria. However, it is always a good idea to double check your steps and calculations to ensure accuracy. Perhaps trying out a few sample numbers and going through the process step by step can help verify your method. Additionally, you can also try using other methods such as listing out all possible combinations and see if you arrive at the same answer to confirm your approach.

Best regards,
 

1. How do you define a four digit number?

A four digit number is a number that contains four digits, with the first digit being non-zero. For example, 1234, 5678, and 9999 are all four digit numbers.

2. What does it mean to have a "property" in combinatorics?

In combinatorics, a "property" refers to a specific characteristic or attribute that a set of numbers or objects possesses. In the context of finding four digit numbers, a property could refer to a certain pattern or arrangement of the digits in the number.

3. How many four digit numbers are there in total?

There are a total of 9,000 four digit numbers, ranging from 1000 to 9999.

4. Can you give an example of a four digit number with a specific property?

Sure, let's say we want to find all four digit numbers where the first two digits are the same and the last two digits are also the same. In this case, an example would be 2299.

5. How do you use combinatorics to find all four digit numbers with a certain property?

In combinatorics, we use principles such as permutations, combinations, and the fundamental counting principle to systematically list out all possible combinations and arrangements of digits in a number. By identifying the specific property we are looking for, we can narrow down the list of four digit numbers and determine which ones have the desired property.

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