How many 4-digit numbers can be formed with a limited number of digits?

In summary, all 4 digit numbers can contain the same digit, any 3 out of the 4 places can be occupied by the same digit and any 2 out of the 4 places can be occupied by the same digit.
  • #1
xiphoid
57
0

Homework Statement


how many 4 digit numbers are there which do not contain more than 2 different numbers?


Homework Equations






The Attempt at a Solution


all can contain the same digit
any 3 out of the 4 places can be occupied by the same digit
and
any 2 out of the 4 places can be occupied by the same digit
 
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  • #2
xiphoid said:

Homework Statement


how many 4 digit numbers are there which do not contain more than 2 different numbers?


Homework Equations






The Attempt at a Solution


all can contain the same digit
any 3 out of the 4 places can be occupied by the same digit
and
any 2 out of the 4 places can be occupied by the same digit


You are on the right track, think about it this way. You have a choice of choosing the numbers from 1 to 9 and then you have three cases:

1. the case where all 4 numbers are the same number as you chose

2. the case where 3 numbers are the same number as you chose as well as you have to choose another number that's different from the other three

3. the case where 2 numbers are the same number as you chose and you have to choose 2 other numbers where they are different from each other and different from the other 2 you previously chose

Does that make sense?
 
  • #3
leej72 said:
You have a choice of choosing the numbers from 1 to 9 and then you have three cases.
Shouldn't that be digits from 0 to 9 (a total of 10 possible digits)?
 
  • #4
rcgldr said:
Shouldn't that be digits from 0 to 9 (a total of 10 possible digits)?

Probably yes, but (presumably) the first (left-most) digit cannot be 0. (Since this is not made clear, if I were doing the problem I would solve both versions.)

RGV
 
  • #5
As you mention, it's not clear if leading zeroes are allowed, like a combination lock with 4 digits. I would assume they are allowed, since otherwise it eliminates all the combinations such as 0xxx, 00xx, 000x, 0000.
 
  • #6
rcgldr said:
Shouldn't that be digits from 0 to 9 (a total of 10 possible digits)?

Yes that was a typo, I did mean from 0 to 9 but I am not too sure if the first number would be allowed to be a 0. I would personally think otherwise, not letting the first number to 0
 
  • #7
as per the answer given, you will need to consider both the cases!
 

What is the difference between permutation and combination?

Permutation and combination are both methods of arranging a set of objects, but they differ in the way they handle repetition. Permutation is used when the order of arrangement matters and does not allow for repetition, while combination is used when the order does not matter and repetition is allowed.

How do I calculate the number of possible permutations or combinations?

The number of permutations or combinations can be calculated using specific formulas. For permutations, the formula is n! / (n-r)! where n is the number of objects and r is the number of objects being arranged. For combinations, the formula is n! / (r! * (n-r)!). It is important to make sure that the objects chosen for the arrangement or combination are distinct.

What is the difference between a permutation with repetition and a combination with repetition?

A permutation with repetition is a method of arranging a set of objects where repetition is allowed. This means that an object can be used more than once in the arrangement. A combination with repetition is a method of selecting a subset from a larger set of objects, where repetition is allowed. This means that an object can be selected multiple times in the subset.

When should I use permutation and when should I use combination?

You should use permutation when the order of arrangement matters and repetition is not allowed. For example, if you are selecting a team of 5 members from a group of 10, the order in which the members are chosen matters. You should use combination when the order does not matter and repetition is allowed. For example, if you are selecting 3 toppings for a pizza from a list of 5 toppings, the order in which the toppings are selected does not matter.

Can permutation and combination be applied in real-life situations?

Yes, permutation and combination can be applied in many real-life situations. For example, it can be used in probability and statistics to calculate the likelihood of events occurring. It can also be used in business decisions, such as determining the number of possible outcomes for a marketing campaign. Additionally, it can be used in everyday activities, such as selecting outfits from a wardrobe or arranging items on a shelf.

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