Rahul@gurome wrote:Thanks jaxis for pointing out the difference.
Here the gifts are different hence there is exist a much more intuitive and easy method as gmat1011 explained.

gmat1011 wrote:Quick question for Rahul: In your stick method ---

If I put 16 different sticks (each standing for a gift) + 3 separators

Then why doesn't (19!)/3!*1! yield the same result... If the gifts are all different then its like dividing by 1! 16 times in the denominator, right? Just as you divide by 16! if the items are all identical. Why doesn't that work here?

Let's analyze for a small number of gifts, say 3 and 2 children.
Say the 4 gifts are A, B, and C and the children are x and y.

Now according to simple application of "stick-separator" method, there are four possible combinations: (0, 3), (1, 2), (2, 1), and (3, 0).

But as the gifts are different, (1, 2) can have three possible sub-arrangements : (A, B + C), (B, A + C), and (C, A + B). Same goes for (2, 1). Hence a total of 8 different combinations.

Whereas if you try to modify the "stick-separator" method as you did, then the number is much more because it takes some false sub-arrangements into account. Such as it interprets (0, 3) as (0, A + B + C), (0, A + C + B), (0, B + A + C), (0, B + C + A), (0, C + A + B), and (0, C + B + A).

Hope this helps to clear the confusion.

Rahul@gurome wrote:

DarkKnight wrote:I can't visualize how we got to 4^16???? I don't understand what to make out of "I think it can be thought of as assigning the children to each distinct gift and not the other way around. " statement. Can you please explain how to get to 4^16? Thanks for your help.

We have 16 different gifts and we have to distribute them among 4 children.

Now, each of the 16 gifts can be given to any one of the 4 children.
Hence each gift can be distributed in 4 ways.
Thus, 16 gifts can be distributed by 4^16 ways.

Hope it helps.

yellowho wrote:In how many ways can 16 different gifts be divided among four children if there is no restriction on the number of gifts that each child can receive? In other words, a child can receive as few as zero gifts and as many as 16 gifts?

gmatdriller wrote:What is the difference if the question had asked:

In how many ways can 4 children be assigned 16 different gifts...?
Assume a child can receive as few as 0 gifts and as many as 16.

Thanks

GMATGuruNY wrote:

gmatdriller wrote:What is the difference if the question had asked:

In how many ways can 4 children be assigned 16 different gifts...?
Assume a child can receive as few as 0 gifts and as many as 16.

Thanks

Same problem.
Since it is possible for a child not to receive a gift, but each gift MUST be given to a child, we count from the perspective of the gifts:
Number of options for the 1st gift = 4. (Any of the 4 children.)
Number of options for the 2nd gift = 4. (Any of the 4 children.)
Number of options for the 3rd gift = 4. (Any of the 4 children.)
Until...
Number of options for the 16th gift = 4. (Any of the 4 children.)
To combine these options, we multiply:
4*4*4*...*4*4*4 = 4¹�.