Question

In how many ways can 6 different books be distributed among 3 students such that none gets equal number of books ?

Answer 1

Number of ways without any constraint $=6!$

$=720$

Now for equal distribution:

1st one will get $2$ books from $6={}^6{C}_2.$

2nd one will get $2$ books from remaining $4={}^4{C}_2.$

3rd one $={}^2{C}_2.$

So, total ways ${}^6{C}_2\times {}^4{C}_2\times {}^2{C}_2$

$=15\times 6\times 1$

$=90$

So the number of ways for none to get equal books is $720-90=630$