In how many ways can a person post 5 letters in 3 letter boxes?

Question

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

gurmukh · Accepted Answer

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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raghav2512 · Answer

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

~~~
Thanks

CrackverbalGMAT · Answer

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

Arun Kumar

MathRevolution · Answer

1st letter has 3 letterboxes choice. Similarly, every letter has 3 choices

=> Therefore, (3*3*3*3*3) =  3^5

Answer D

KumarSurbhit · Answer

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 answer

Or say this way
to select first letterbox among the three = 3C1
 max all 5 letter can get into it = 5
so, for first = 3C1*5
likewise for second = 2C1 *4
for third = 3
Answer = 3C1*5 + 2C1*4 + 3 = 26

CrackverbalGMAT · Answer

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 helps

Arun Kumar

ScottTargetTestPrep · Answer

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

Karpov92 · Answer

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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LSwetha · Answer

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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IanStewart · Answer

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?

Fdambro294 · Answer

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.

IanStewart · Answer

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.

Kinshook · Answer

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

raghav2512 · Answer

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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Fdambro294 · Answer

Thank you much.

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bhavika01 · Answer

3^5!

A very basic yet interesting question.

luisdicampo · Answer

Deconstructing the Question  
A person posts  5 letters  into  3 letterboxes .
Each letter can go into any of the 3 boxes (independent choices).
Key idea: for  n distinct items  each with  k choices , total ways =  k^n .

Step-by-step

Each of the 5 letters has 3 choices:
 3\cdot3\cdot3\cdot3\cdot3=3^5

Answer: 3^5

In how many ways can a person post 5 letters in 3 letter boxes?

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