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

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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