$\color{red}{Distinguishable \ Objects \ and \ Indistinguishable \ Boxes - }$
It's similar to the problem of - Number of ways to partition a set of r objects into n non-empty subsets.
It is given by $\color{green}{Stirling \ Number \ of \ 2^{nd} \ kind}- \color{magenta}{S(r,n)} $
Recurrence Equation - $S(r+1,n)=S(r,n-1)+ n.S(r,n)$
For the above question - r =6 and n = 4.
S(6,4) -
1
1 1
1 3 1
1 7 6 1
1 15 25 10 1
1 31 90 $\color{Orange}{65}$ 15 1
$\color{DarkBlue}{Note} - 65$ is obtained as $25 \ +\ 4*(10.)$ In the similar way, all the numbers are obtained.
For more detail - Refer this
$\color{red}{Indistinguishable \ Objects \ and \ Indistinguishable \ Boxes - }$
Refer this.