Problem Solving

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

(A) 480
(B) 1024
(C) 54
(D) 3^5
(E) 5^3

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

(A) 480
(B) 1024
(C) 54
(D) 3^5
(E) 5^3

Apparently, the solution is quite simple here:
each letter will have 3 boxes to choose from and so answer 3^5.

But I am having a tough time visualizing this solution. The way I am looking at is:


1box 2box 3box
5 0 0
4 0 1
3 0 2
3 1 1
2 2 1

Now these can be arranged internally for example - 500 could be 005 etc.

What mistake am I making? Thanks

mensanumber wrote:

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

(A) 480
(B) 1024
(C) 54
(D) 3^5
(E) 5^3

Apparently, the solution is quite simple here:
each letter will have 3 boxes to choose from and so answer 3^5.

But I am having a tough time visualizing this solution. The way I am looking at is:


1box 2box 3box
5 0 0
4 0 1
3 0 2
3 1 1
2 2 1

Now these can be arranged internally for example - 500 could be 005 etc.

What mistake am I making? Thanks

Not making any mistakes so far but haven't completed the thought.

As you suggest, 500 can be arranged 3 ways.

Likewise, 401 can be arranged 3 ways.

And so on. All together there are then 3x3x3x3x3 = 3^5 ways

Brent@GMATPrepNow wrote:

mensanumber wrote:

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

(A) 480
(B) 1024
(C) 54
(D) 3^5
(E) 5^3

Apparently, the solution is quite simple here:
each letter will have 3 boxes to choose from and so answer 3^5.

But I am having a tough time visualizing this solution. The way I am looking at is:


1box 2box 3box
5 0 0
4 0 1
3 0 2
3 1 1
2 2 1

Now these can be arranged internally for example - 500 could be 005 etc.

What mistake am I making? Thanks

If that's your final solution, then you appear to be treating the 5 letters as 5 IDENTICAL letters, and you are treating the 3 mailboxes as IDENTICAL mailboxes

Cheers,
Brent

Hi Brent,
Thanks for your reply.

That's not my full solutions. Here it is:

1box 2box 3box
5 0 0......................................Case-1
4 0 1......................................Case-2
3 0 2......................................Case-3
3 1 1......................................Case-4
2 2 1......................................Case-5

Let's consider each case individually,

Case-1
500 i.e. 5 letters in box1, 0 letters in box-2, 0 letters box-3. Now, there will be 3 such scenarios, 3P1/2P1 = 3. (by MISSISSIPPI rule dividing by 2P1 for two identical zeros) These 3 scenarios are 500, 050, 005
Since letters distinct, for each of these scenarios: 5C5*0C0*0C0
So, total possible arrangements for case-1 : (3P1/2P1)*5C5*0C0*0C0 = 3*1*1*1 = 3

Similarly,
Case-2: 401, total possible arrangements : (3P1)*5C4*1C0*1C1 = 6*5*1*1 = 30
Case-3: 302, total possible arrangements : (3P1)*5C3*2C0*2C2 = 6*10*1*1 = 60
Case-4: 311, total possible arrangements : (3P1/2P1)*5C3*2C1*1C1 = 3*10*2*1 = 60
Case-4: 221, total possible arrangements : (3P1/2P1)*5C2*3C2*1C1 = 3*10*3*1 = 90

Total of all cases = 3+30+60+60+90 = 243 which is indeed 3^5

But what irritates me is my long winded solution is visualize-able to me and not your more elegant solution.
Can you help me see it from your point of view?

Thanks again

regor60 wrote:

mensanumber wrote:

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

(A) 480
(B) 1024
(C) 54
(D) 3^5
(E) 5^3

Apparently, the solution is quite simple here:
each letter will have 3 boxes to choose from and so answer 3^5.

But I am having a tough time visualizing this solution. The way I am looking at is:


1box 2box 3box
5 0 0
4 0 1
3 0 2
3 1 1
2 2 1

Now these can be arranged internally for example - 500 could be 005 etc.

What mistake am I making? Thanks

Not making any mistakes so far but haven't completed the thought.

As you suggest, 500 can be arranged 3 ways.

Likewise, 401 can be arranged 3 ways.

And so on. All together there are then 3x3x3x3x3 = 3^5 ways

Thanks for your reply Regor.

Here is my full solution:

1box 2box 3box
5 0 0......................................Case-1
4 0 1......................................Case-2
3 0 2......................................Case-3
3 1 1......................................Case-4
2 2 1......................................Case-5

Let's consider each case individually,

Case-1
500 i.e. 5 letters in box1, 0 letters in box-2, 0 letters box-3. Now, there will be 3 such scenarios, 3P1/2P1 = 3. (by MISSISSIPPI rule dividing by 2P1 for two identical zeros) These 3 scenarios are 500, 050, 005
Since letters distinct, for each of these scenarios: 5C5*0C0*0C0
So, total possible arrangements for case-1 : (3P1/2P1)*5C5*0C0*0C0 = 3*1*1*1 = 3

Similarly,
Case-2: 401, total possible arrangements : (3P1)*5C4*1C0*1C1 = 6*5*1*1 = 30
Case-3: 302, total possible arrangements : (3P1)*5C3*2C0*2C2 = 6*10*1*1 = 60
Case-4: 311, total possible arrangements : (3P1/2P1)*5C3*2C1*1C1 = 3*10*2*1 = 60
Case-4: 221, total possible arrangements : (3P1/2P1)*5C2*3C2*1C1 = 3*10*3*1 = 90

Total of all cases = 3+30+60+60+90 = 243 which is indeed 3^5

But what irritates me is my long winded solution is visualize-able to me and not your more elegant solution.
Can you help me see it from your point of view?

Thanks again

mensanumber wrote:

regor60 wrote:

mensanumber wrote:

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

(A) 480
(B) 1024
(C) 54
(D) 3^5
(E) 5^3

Apparently, the solution is quite simple here:
each letter will have 3 boxes to choose from and so answer 3^5.

But I am having a tough time visualizing this solution. The way I am looking at is:


1box 2box 3box
5 0 0
4 0 1
3 0 2
3 1 1
2 2 1

Now these can be arranged internally for example - 500 could be 005 etc.

What mistake am I making? Thanks

Not making any mistakes so far but haven't completed the thought.

As you suggest, 500 can be arranged 3 ways.

Likewise, 401 can be arranged 3 ways.

And so on. All together there are then 3x3x3x3x3 = 3^5 ways

Thanks for your reply Regor.

Here is my full solution:

1box 2box 3box
5 0 0......................................Case-1
4 0 1......................................Case-2
3 0 2......................................Case-3
3 1 1......................................Case-4
2 2 1......................................Case-5

Let's consider each case individually,

Case-1
500 i.e. 5 letters in box1, 0 letters in box-2, 0 letters box-3. Now, there will be 3 such scenarios, 3P1/2P1 = 3. (by MISSISSIPPI rule dividing by 2P1 for two identical zeros) These 3 scenarios are 500, 050, 005
Since letters distinct, for each of these scenarios: 5C5*0C0*0C0
So, total possible arrangements for case-1 : (3P1/2P1)*5C5*0C0*0C0 = 3*1*1*1 = 3

Similarly,
Case-2: 401, total possible arrangements : (3P1)*5C4*1C0*1C1 = 6*5*1*1 = 30
Case-3: 302, total possible arrangements : (3P1)*5C3*2C0*2C2 = 6*10*1*1 = 60
Case-4: 311, total possible arrangements : (3P1/2P1)*5C3*2C1*1C1 = 3*10*2*1 = 60
Case-4: 221, total possible arrangements : (3P1/2P1)*5C2*3C2*1C1 = 3*10*3*1 = 90

Total of all cases = 3+30+60+60+90 = 243 which is indeed 3^5

But what irritates me is my long winded solution is visualize-able to me and not your more elegant solution.
Can you help me see it from your point of view?

Thanks again

I read your response too quickly so ignore what I wrote, it's not correct. The way you've done it is correct.

Each letter has 3 options.
So, total no of ways = 3 x 3 x 3 x 3 x 3 = 3^5