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This question is same as solving 6 balls and unleveled container problem

Now, we can arrange those $6$ numbers in $4$ container as $2$ balls and $2$ balls in first two container and $1$ ball and $1$ ball in last two container

Case $1:$

If we take $1$ st box contain $\left \{ 1,2 \right \}$ and $2$ nd box contain $\left \{ 3,4 \right \}$

$3$rd and $4$ th box contains $\left \{ 5 \right \},\left \{ 6 \right \}$

Then, total number of selection possible $\binom{6}{2}\times \binom{4}{2}\times \binom{2}{1}\times \binom{1}{1}$

but here $\left \{ 5 \right \},\left \{ 6 \right \}$ can arrange among themselves (here we need to divide by $2!$)

and $\left \{ 1,2 \right \}$ and $\left \{ 3,4 \right \}$ can arrange among themselves, which are no need to be counted (here we need to divide by $2!$)

So, for $2,2,1,1$ total number of arrangement possible $\frac{\binom{6}{2}\times \binom{4}{2}\times \binom{2}{1}}{2!2!}=45$

Case $2:$

All box contain $2$ balls

And boxes are indistinguishable

So, total no. of arrangement possible $\frac{\binom{6}{2}\times \binom{4}{2}\times \binom{2}{2}}{3!}=15$

Note: Here we are doing arrangement of container , and not balls

So, from $(1)$ and $(2)$ we get $45+15=60$