How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?

The answer to this is 3^5

Explanation: 1st Letter can be posted in any of the 3 mailboxes, 2nd letter can also be posted in any of the 3 Mailboxes and so on...

so, total possible ways=3*3*3*3*3

I understood this explanation, but;

My Doubt: Why is the answer not 5^3 ? I mean, any of the 5 letters can be posted in 1st Mailbox, similarly, any of the 5 letters can be posted in 2nd Mailbox, same goes for 3rd Mailbox.

so, total possible ways= 5*5*5

Where am I going wrong?

For the given case: Find the total no. of ways in which a 3 digit number can be formed with the digits- 2,4,9,8,5 , given repetition of digits is allowed.

Here, the unit's place has 5 choice of numbers, similarly the tens place has 5 choices and so on.

so, total ways=5*5*5

If this is correct, then why in the above original question, this method or logic is not available?

I am new to this, please explain where I am going wrong? What i am missing?

NOTE: I know this question is already asked in the community before, but my doubt , which I have discussed, is not addressed anywhere, Please dont mark as DUPLICATE

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How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?

In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  04 Jun 2020, 12:56

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?

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How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
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How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
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In how many ways can a person post 5 letters in 3 letterboxes?(A) 480(B) 1024(C) 54(D) 3^5

(E) 5^3

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How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?

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Re: In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  04 Jun 2020, 16:45

For first letter there are 3 choices, similarly for all rest of the letters also there are 3 choices hence total number of choices are 3×3×3×3×3 = 3^5.

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In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  19 Aug 2020, 18:45

The rule of arrangements say that the number of ways of arranging/distributing n distinct things into r different things is given as \(r^n\). Here letters go into the post boxes. Therefore post boxes = n = 5 and letters = r = 3\(r^n\) = \(3^5\) = 81

Option D

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Re: In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  19 Aug 2020, 23:20

1st letter has 3 letterboxes choice. Similarly, every letter has 3 choices => Therefore, (3*3*3*3*3) = \(3^5\)

Answer D

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In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  Updated on: 30 Aug 2020, 09:43

Great question. This is how I would approach this.Let's understand this in a simple way without using any formulas-We know that in this question no restrictions have been given.

Therefore, each letter has 3 choices. It can either go to Letter Box 1, 2 ,or 3.

We have 5 letters. So, the number of ways become-

3 * 3 * 3 * 3 * 3 = \(3^5 \)

Option D.

Let's try to analyze a few more scenarios:


1) In how many ways can a person post 3 out of 5 distinct letters in 3 letter-boxes such that each letter box can hold only 1 letter?
2) In how many ways can a person post 5 letters in 3 letter-boxes if all the letters are not posted in the same letter box?
.....

Solution:


1) In the first scenario we have a restriction that each box can have only 1 letter, which essentially means that the number of choices for each letter box are but limited. _ _ _We find that we have 5 choices for the first letter box. After filling the first, we have 4 choices left for the second and 3 for the last.

So, the choices become 5 * 4 * 3 = 60 ways

(Note: This scenario is the same as arranging 5 letters in 3 places or 5P3)We can also do it by choosing 3 letters out of 5 in 5C3 ways and then arranging them in 3! ways= 5C3 * 3! = 60

2) The second one is another restrictive case.


We already know that the number of ways a person can post 5 letters in 3 letter-boxes are \(3^5\). Let's find the number of ways in which all letters can be posted in the same letter box and subtract from the total. Hence, the number of ways in which this can be achieved is 3 (as we can post all the 5 letters in a single letter box in 3 ways). So, the number of ways a person can post 5 letters in 3 letter-boxes if all the letters are not posted in the same letter

box are (\(3^5\) -3) = 240

~~~

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How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?

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Originally posted by raghav2512 on 20 Aug 2020, 00:42.
Last edited by raghav2512 on 30 Aug 2020, 09:43, edited 1 time in total.

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Re: In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  20 Aug 2020, 13:30

davidbeckham wrote:

In how many ways can a person post 5 letters in 3 letterboxes?(A) 480(B) 1024(C) 54(D) 3^5

(E) 5^3

Can someone please help address my doubt. I agree my answer is not correct but can anyone please point out where am I going wrong.What if I try to solve it based on letterboxes such that each of the letterbox can receive max 5 letters.hence, 5*5*5 = 5^3 is the answerOr say this wayto select first letterbox among the three = 3C1 max all 5 letter can get into it = 5so, for first = 3C1*5likewise for second = 2C1 *4for third = 3

Answer = 3C1*5 + 2C1*4 + 3 = 26

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In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  20 Aug 2020, 16:56

KumarSurbhit wrote:

davidbeckham wrote:

In how many ways can a person post 5 letters in 3 letterboxes?(A) 480(B) 1024(C) 54(D) 3^5

(E) 5^3

Can someone please help address my doubt. I agree my answer is not correct but can anyone please point out where am I going wrong.What if I try to solve it based on letterboxes such that each of the letterbox can receive max 5 letters.hence, 5*5*5 = 5^3 is the answerOr say this wayto select first letterbox among the three = 3C1 max all 5 letter can get into it = 5so, for first = 3C1*5likewise for second = 2C1 *4for third = 3

Answer = 3C1*5 + 2C1*4 + 3 = 26

Hello Kumar Surbhit. Yours is a doubt which is very valid. 'Why can't it be done this way?'The simplest way in these type of questions is think, who is going into what. Are letters going into the post boxes or post boxes going into the letters. Here it becomes clear that letter will go into the post boxes. So each letter can go into any of the 3 post boxes and therefore letter 1, 2, 3 4 and 5 each have 3 options = 3 * 3 * 3 * 3 * 3 = 81.Take another example of 4 friends who want to stay in 5 hotels. is the answer \(4^5\) or \(5^4\).Lets understands this. Will the friends go to the hotels or the hotels go to the friends. It would be friends going to the hotels. So the first friend can stay in any of the 5 hotels, the 2nd too can stay in any 5 and so can the 3rd and 4th person.Therefore the answer is \(5 * 5 * 5 * 5 = 5^4\)Hope this helpsArun Kumar _________________

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Re: In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  24 Aug 2020, 09:48

davidbeckham wrote:

In how many ways can a person post 5 letters in 3 letterboxes?(A) 480(B) 1024(C) 54(D) 3^5

(E) 5^3


Solution:The first letter has 3 choices of letterboxes to be placed in, as do the second, third, fourth, and the fifth letters. Therefore, the total number of ways the 5 letters can be placed into the 3 letterboxes is 3 x 3 x 3 x 3 x 3 = 3^5.

Answer: D

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In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  24 Aug 2020, 10:16

To get rid of the confusion whether the ans is 5^3 or 3^5,I use this approach-Let's say letters are L1,L2,L3 and Boxes are B1,B2,B3,B4,B5.1.Can a possible combination be L1L1L1? No,1 letter can't be in 3 boxes simultaneously.so 5*5*5*5*5 cant be true.2. Can a possible combination be B1B1B1B1B1?Yes cz 5 letters can be in a single box.So 3*3*3 is the ans.(D)

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Re: In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  25 Aug 2020, 05:41

Each letter has the choice of going to any box and the operation between two letters is and therefore 3^5

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Re: In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  25 Aug 2020, 07:32

raghav2512 wrote:

Let's try to analyze a few more scenarios:

1) In how many ways can a person post 5 letters in 3 letter-boxes such that each letter box can hold only 1 letter?

[b]Solution:


1) In the first scenario we have a restriction that each box can have only 1 letter, which essentially means that the number of choices for each letter box are but limited. _ _ _We find that we have 5 choices for the first letter box. After filling the first, we have 4 choices left for the second and 3 for the last.

So, the choices become 5 * 4 * 3 = 60 ways


It can be useful to consider variations on questions you study, but in this particular example, you've solved a different question than the one you asked. The question you started from was: In how many ways can a person post 5 letters in 3 letter-boxes such that each letter box can hold only 1 letter? The answer to that question is zero. If each of the 3 boxes can only hold 1 letter, they can hold at most 3 letters in total, so you can't put 5 letters in them.

The question you solved (correctly) instead was this one, with the '3' and '5' reversed: In how many ways can a person post 3 letters in 5 letter-boxes such that each letter box can hold at most 1 letter?

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How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?

Re: In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  29 Aug 2020, 20:49

IanStewart wrote:

raghav2512 wrote:

Let's try to analyze a few more scenarios:

1) In how many ways can a person post 5 letters in 3 letter-boxes such that each letter box can hold only 1 letter?

[b]Solution:


1) In the first scenario we have a restriction that each box can have only 1 letter, which essentially means that the number of choices for each letter box are but limited. _ _ _We find that we have 5 choices for the first letter box. After filling the first, we have 4 choices left for the second and 3 for the last.

So, the choices become 5 * 4 * 3 = 60 ways


It can be useful to consider variations on questions you study, but in this particular example, you've solved a different question than the one you asked. The question you started from was: In how many ways can a person post 5 letters in 3 letter-boxes such that each letter box can hold only 1 letter? The answer to that question is zero. If each of the 3 boxes can only hold 1 letter, they can hold at most 3 letters in total, so you can't put 5 letters in them.

The question you solved (correctly) instead was this one, with the '3' and '5' reversed: In how many ways can a person post 3 letters in 5 letter-boxes such that each letter box can hold at most 1 letter?

Ian,If the question were written as the following:"In How many ways can 3 Letters out of 5 Distinct Letters be posted in 3 Letter Boxes such that each Letter Box can hold only 1 Letter?"Would that be the correct way to form the question?

Because then you 1st have to decide how many Combinations of 3 Letters we can have that end up going in to the 3 mailboxes. Then 2nd, for each 1 of these Combinations, you need to figure out the different possible Arrangements of the 3 Letters in the 3 Boxes.

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Re: In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  30 Aug 2020, 03:51

Fdambro294 wrote:

Ian,If the question were written as the following:"In How many ways can 3 Letters out of 5 Distinct Letters be posted in 3 Letter Boxes such that each Letter Box can hold only 1 Letter?"Would that be the correct way to form the question?

Because then you 1st have to decide how many Combinations of 3 Letters we can have that end up going in to the 3 mailboxes. Then 2nd, for each 1 of these Combinations, you need to figure out the different possible Arrangements of the 3 Letters in the 3 Boxes.

Yes, if you did rephrase raghav's question in that way, it would then make sense, and that's probably the question raghav intended. And it would be correct to answer that question the way you suggest - pick the set of envelopes first, then work out how many choices you have for each envelope you've picked. _________________

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Re: In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  30 Aug 2020, 07:49

Asked: In how many ways can a person post 5 letters in 3 letterboxes?For each letter there are 3 options (letterboxes). Total number of ways = 3^5 IMO D _________________

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In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  30 Aug 2020, 09:35

IanStewart wrote:

Fdambro294 wrote:

Ian,If the question were written as the following:"In How many ways can 3 Letters out of 5 Distinct Letters be posted in 3 Letter Boxes such that each Letter Box can hold only 1 Letter?"Would that be the correct way to form the question?

Because then you 1st have to decide how many Combinations of 3 Letters we can have that end up going in to the 3 mailboxes. Then 2nd, for each 1 of these Combinations, you need to figure out the different possible Arrangements of the 3 Letters in the 3 Boxes.

Yes, if you did rephrase raghav's question in that way, it would then make sense, and that's probably the question raghav intended. And it would be correct to answer that question the way you suggest - pick the set of envelopes first, then work out how many choices you have for each envelope you've picked.

That's exactly the question I intended Ian. Will correct it. Then in that case we can choose 3 out of 5 letters in 5C3 ways and arrange these 3 letters in 3 boxes in 3! ways as Fdambro mentioned i.e. 5C3 * 3! or via permutation 5*4*3. Thanks for pointing out the mistake.

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Re: In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  31 Aug 2020, 05:06

Thank you much.

IanStewart wrote:

Fdambro294 wrote:

Ian,If the question were written as the following:"In How many ways can 3 Letters out of 5 Distinct Letters be posted in 3 Letter Boxes such that each Letter Box can hold only 1 Letter?"Would that be the correct way to form the question?

Because then you 1st have to decide how many Combinations of 3 Letters we can have that end up going in to the 3 mailboxes. Then 2nd, for each 1 of these Combinations, you need to figure out the different possible Arrangements of the 3 Letters in the 3 Boxes.

Yes, if you did rephrase raghav's question in that way, it would then make sense, and that's probably the question raghav intended. And it would be correct to answer that question the way you suggest - pick the set of envelopes first, then work out how many choices you have for each envelope you've picked.

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Re: In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?
  13 Sep 2021, 12:17

3^5!

A very basic yet interesting question.

How many ways can 5 letters be posted in 3 post boxes if any number of letters can be posted in all of the three post?

Re: In how many ways can a person post 5 letters in 3 letter boxes? [#permalink]