In how many ways 5 different balls can be distributed into 3 boxes so that no box remains empty?

Explanation

The required number of ways
`=3^(5)-.^(3)C_(1)(3-1)^(5)+.^(3)C_(2)(3-2)^(5)-.^(3)C_(3)(3-3)^(5)`
`=243-96+3-0=150`
Or Coefficient of `x^(5)` inn `5!(e^(x)-1)^(3)`
=Coefficient of `x^(5)` in `5!(e^(3x)-3e^(2x)+3e^(x)-1)`
`=5!((3^(5))/(5!)-3xx(2^(5))/(5!)+3xx(1)/(5!))=3^(5)-3*2^(5)+3=243-96+3=150`