Combinatorics Problem: Sending 15 Postcards to 15 Friends in Unique Ways

[swtlilsoni]

Homework Statement

You have 3 types of postcards. There are 5 of each type. How many ways can you send the 15 postcards to 15 friends, if each friend receives 1.

The Attempt at a Solution

I thought it would merely be 15!/(3*5!)

 

[CompuChip]

Are you sure about the 3 * 5! ?

 

[swtlilsoni]

Are you sure about the 3 * 5! ?

because there are three sets of five identicals

 

[tiny-tim]

hi swtlilsoni!

because there are three sets of five identicals

if there were 10 friends, and 2 sets of five identicals, would you use 10!/2*5! ?

 

[swtlilsoni]

Ohh okay so it would be 5!³!

 

[CompuChip]

Can you explain that to us, or are you just guessing now? :)

 

[swtlilsoni]

it's because it has to be multiplied. For every rearrangement of five identicals, there are two more rearrangements of the others

 

[tiny-tim]

hi swtlilsoni!

(just got up :zzz: …)

it's because it has to be multiplied. For every rearrangement of five identicals, there are two more rearrangements of the others

the general rule for selecting a of one type, b of another, … z of another, from n altogether (with a+b+… +z = n), is:

n!/a!b!…z!​

for only two types, that reduces to the familiar:

n!/a!b! = n!/a!(n-a)! = ⁿC_a ​