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the number of ways for distributing "n" different things in to "r" different groups so that no group is left blank is given by ? and what is the proof. please please say me this iitians

the number of ways for distributing "n" different things in to "r" different groups so that no group is left blank is given by
?
and what is the proof.
please please say me this iitians

Grade:12th pass

1 Answers

jitender lakhanpal
62 Points
9 years ago

Dear Nikhil,

there can be three cases when n<r then the number of ways are 0 as there will be atleast one group that will be vacant

when n = r 

then the number of ways are n factorial

when n>r then

the number of ways would be P(n,r) as there would be atleast one object in one group and then remaining n-r objects have to be placed in r things now there would be further 2 cases as

case:1 n-r>r   then the total number of cases would be P(n,r)*P((n-r),r)  

case:2 n-r<r   then the total number of cases would be P(n,r)*[(C(n-r,1)*r + C(n-r,2)*(r-1)*2fact + C(n-r,3)*(r-2)*3fact-------

C(n-r,n-r)*(2r-n+1)*(n-r)fact       

we got this expression by first arranging n objects in r places so we got P(n,r) AND then from remaining n-r objects we select 1 object and arrange in r places we got C(n-r,1)*r OR select 2 objects then arrange them in r places OR select 3 objects then arrange them in r places this will go upto n-r objects.

and we know that AND means multiplication OR means ADDITION

by this we will get the arrangement when no group will be vacant.

 

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Jitender

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