You want to send postcards to 12 friends. In the shop, there are only 3 kinds of

Question

You want to send postcards to 12 friends. In the shop, there are only 3 kinds of postcards. In how many ways can you send the postcards if (a) there is a large number of each kind of postcards, and you want to send one card for each friend. (b) there is a large number of each kind of postcards, you are willing to and send one or more postcards to each friend, but no one should get two identical postcards. (c) the shot has only 4 of each kind of postcard, and you want to send one card to each friend.

Explanation / Answer

(a)
for every friend you can select any of the postcard out of 3 different kinds. Hence this selection can be done in 3C1 = 3 ways

Hence number of ways this can be achieved is 3^12 = 531,441

(b)
For every friend you can select either 1, 2 or 3 kinds of cards.
This can be done in 3C1 + 3C2 + 3C3 = 3 + 3 + 1 = 7 ways

hence number of ways this can be achieved is 7^12 = 13,841,287,201

(c)
If there are only 4 cards available of each kind, this means there are 12 cards available and we need to send it to 12 friends.

You can send 4 cards of one type to any of 4 friends, this can be done in 3C1*12C4 ways
Now, 4 cards of another type can be sent to any of 4 friends from the remaining 8 friends in 2C1*8C4 ways
Finally, 4 cards and 4 friends, this can be done in only 1 ways

Hence total number of ways to achieve this is 3C1 * 12C4 * 2C1 * 8C4 * 1 = 207,900

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