a basic combination -- help needed

[smanstar]

Q
In how many ways can 5 letters be posted in 4 mail boxes , any letter can be posted in any mail box ??

How i tried to solve

5 letters each letter has 4 choices so 4*4*4*4*4 = 4^5 ways

but I get confused, when I tried to solve the same in this way

4 mail boxes each can accept 5 letters so 5*5*5*5 = 5^4 ways.

please help me out with the basic difference between the two approaches and secondly how to solve the same ??

[GMATGuruNY]

Q
In how many ways can 5 letters be posted in 4 mail boxes , any letter can be posted in any mail box ??

How i tried to solve

5 letters each letter has 4 choices so 4*4*4*4*4 = 4^5 ways

but I get confused, when I tried to solve the same in this way

4 mail boxes each can accept 5 letters so 5*5*5*5 = 5^4 ways.

please help me out with the basic difference between the two approaches and secondly how to solve the same ??

There are fewer mailboxes than letters.
The result: while not every mailbox must contain a letter, EVERY LETTER MUST CHOOSE A MAILBOX.
Thus, we count from the perspective of the LETTERS.
For each of the five letters, there are 4 choices of mailbox:
4*4*4*4*4 = 4�.

[smanstar]

Q
In how many ways can 5 letters be posted in 4 mail boxes , any letter can be posted in any mail box ??

How i tried to solve

5 letters each letter has 4 choices so 4*4*4*4*4 = 4^5 ways

but I get confused, when I tried to solve the same in this way

4 mail boxes each can accept 5 letters so 5*5*5*5 = 5^4 ways.

please help me out with the basic difference between the two approaches and secondly how to solve the same ??

There are fewer mailboxes than letters.
The result: while not every mailbox must contain a letter, EVERY LETTER MUST CHOOSE A MAILBOX.
Thus, we count from the perspective of the LETTERS.
For each of the five letters, there are 4 choices of mailbox:
4*4*4*4*4 = 4�.

thanks for the reply but I have a doubt again

If in a problem we have to distribute 10 coins to 3 children then it should be 3*3*3...*3 = 3^10 but I am not getting the answer .
Can you please also explain whats the difference between the two problems

[GMATGuruNY]

Q
In how many ways can 5 letters be posted in 4 mail boxes , any letter can be posted in any mail box ??

How i tried to solve

5 letters each letter has 4 choices so 4*4*4*4*4 = 4^5 ways

but I get confused, when I tried to solve the same in this way

4 mail boxes each can accept 5 letters so 5*5*5*5 = 5^4 ways.

please help me out with the basic difference between the two approaches and secondly how to solve the same ??

There are fewer mailboxes than letters.
The result: while not every mailbox must contain a letter, EVERY LETTER MUST CHOOSE A MAILBOX.
Thus, we count from the perspective of the LETTERS.
For each of the five letters, there are 4 choices of mailbox:
4*4*4*4*4 = 4�.

thanks for the reply but I have a doubt again

If in a problem we have to distribute 10 coins to 3 children then it should be 3*3*3...*3 = 3^10 but I am not getting the answer .
Can you please also explain whats the difference between the two problems

How many ways can 10 different coins be distributed among 3 children?

Here, the coins are all DIFFERENT.
Thus, it matters whether Coin A goes to child 1, child 2, or child 3.
In this case, the number of options for each coin = 3. (Any of the 3 children.)
To combine the number of options for each coin, we multiply:
3*3*3*3*3*3*3*3*3*3 = 3¹�.

How many ways can 10 identical coins be distributed among 3 children?

Here, the coins are all IDENTICAL.
Thus, it DOESN'T matter whether Coin A goes to child 1, child 2, or child 3.
All that matters is HOW MANY coins each child receives.
For this sort of problem, I recommend the SEPARATOR method.
Check here:

https://www.beatthegmat.com/combinations-t120668.html