Counting sets a How many ways are there to give 1000 identic

(Counting sets)

a) How many ways are there to give 1000 identical wooden clowns to 7 camp counselors?

b) How many ways are there to give 1000 identical wooden clowns to 7 camp counselors if each counselor must get at least 50?

Solution

a)

So we have n=1000 objects to be distribute to r=7 couselors each may have 0 or more clowns

This is done in

C(1000+7-1,7-1)=C(1006,6)

b)

So each gets at least 50 so we can first distribute 50 to each conselor which is done in 1 way since clowns are identical.

So we are left with 1000-7*50=650 clowns

These are to be distributed to 7 counselors each may get 0 or more of these

So , C(650+7-1,7-1)=C(656,6)

Hence number if ways is

C(656,6)