You have $20$ cards and $12$ envelopes(labelled $1.....12$). In how many ways can you put $20$ cards in envelopes if
a) the cards are distinct
b) the cards are identical
c) the cards are identical and no envelope can be left empty.
Please provide me with some briefing on how to solve this problem. Thank you
On
a) for each of the $20$ cards there are $12$ possibilities, so...
b) To be found is the number of sums $c_1+\cdots+c_{12}=20$ where the $c_i$ are nonnegative integers. For this you can apply stars and bars.
c) To be found is the number of sums $c_1+\cdots+c_{12}=20$ where the $c_i$ are positive integers.
Define $c'_i:=c_i-1$ and find the number of sums $c'_1+\cdots+c'_{12}=20-12=8$ where the $c'_i$ are nonnegative integers, again with stars and bars.
On
a) $$ \frac{31!}{11!} $$
A example:
$$ 1, 3, 5 | 2 | 4 | | | 6, 7, 8 | | | | | | 9, 10, 11, 12, ..., 20 $$
31 Symbols (20 numbers + 11 separators). The 11 separators repeating. The separators, separating the envelopes.
b) $$ \frac{31!}{11! \cdot 20!} $$
A example:
$$ A, A, A | A | A | | | A, A, A | | | | | | A, A, A, A, ..., A $$
31 Symbols (20 'A' + 11 separators). The 11 separators repeating. The separators, separating the envelopes.
c) $$ \frac{19!}{11! \cdot 8!} $$
A example:
$$ A, A, A | A | A | A | A | A, A, A | A | A | A | A | A | A, A, A, A, A $$
31 Symbols (20 'A' + 11 separators). The 11 separators repeating. The separators, separating the envelopes.
In this example we take an A from each envelope and it remains:
$$ A, A | | | | | A, A | | | | | | A, A, A, A $$
19 Symbols (8 'A' + 11 separators). The 11 separators repeating. The separators, separating the envelopes.
HINTS:
For a) think how many different places/envelopes can card nr $1$ end up in? What is the situation with card nr $2$? You may found useful the rule of product.
For b) you could read up on stars and bars.
For c) start by fulfilling the condition and put one card in each envelope. How many cards do you have left? Which of a) or b) is to be used with the leftover cards?
Hope this helped