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I have got this problem in my book:

a)How to distribute 5 different gifts among three persons where each person can get multiple gifts?

b) How to distribute 5 different gifts among three persons where each person must get at least 1 gift?

for the first problem i started like this:

i) choosing 3 gifts from 5 gifts (each people will get one gift)

ii) choosing 2 gifts for one person and rest 2 people will get to choose 2 gifts from 3 gifts.

iii)choosing 3 gifts for one person and rest 2 people will get to choose 2 gifts from 3 gifts.

iv)choosing 4 gifts for one person and rest 2 people will get to choose 1 gifts from 3 gifts.

v) choosing 5 gifts for one person and rest 2 people will not get any gift.

But i'm stuck when applying it and i've got no idea how to solve (b). Please help me out.

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    My interpretation would be that each person is unique and each gift is unique. I would further interpret that every gift is received by a person and no gift is left ungiven. For part (a) pick who gets gift#1, pick who gets gift#2, pick who gets gift#3, etc... Apply multiplication principle to see there are $3^5$ arrangements. For part (b) do so similarly but apply inclusion-exclusion on each of the events that a person doesn't receive any gifts. – 2017-02-04
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    If you interpret it that some gifts are allowed to be left ungiven, then approach similarly, but include a fourth option for each place the gift can go. – 2017-02-04
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    @JMoravitz can you please show the calculation of b? – 2017-02-04
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    I'll leave that for you to do, but note that this can be phrased as counting the number of functions from $\{present1,present2,present3,present4,present5\}$ to $\{person1,person2,person3\}$ (or to $\{person1,person2,person3,noone\}$) satisfying certain conditions. This is a well known and understood problem, and appears on the list for the [twelvefold way](https://en.wikipedia.org/wiki/Twelvefold_way). – 2017-02-04
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    @JMoravitz thanks that means i have to find out the number of on-to functions – 2017-02-04
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    yes in the first interpretation where every gift must be given to someone., no in the second interpretation where it is possible for some gifts to be given to noone. In the case that gifts can be given to noone, it isn't quite the set of onto functions since we don't care if there is a gift sent to noone or not. (*but the method of deriving the formula to use is the same*) – 2017-02-04
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    @JMoravitz oh thanks a lot :) – 2017-02-04

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