What is the number of ways to arrange 10 distinct objects into 4 distinct boxes, where each box hold no more than 4 objects
This question was asked earlier today, but has been deleted, I don't know why
I attempted to solve the problem, and came onto this
Put $10$ distinct object in $4$ distinct boxes, where a box cannot hold more than $4$ object
Distinct object say- $(1,2,3,4,5,6,7,8,9,10)$
Distinct boxes say- $(x_1,x_2,x_3,x_4)$
So generally there are $16$ spaces in all box, to which we want to fix in $10$ distinct objects so some spaces might be left
There are two ways of doing this
IF THE OBJECTS ARE NOT REPEATED
If the object are not repeated, there would be $6$ spaces each time
I'll call this the Occurance of $6$ space ( no extra object )
So the number of ways of doing this is $10!×16C10 = 10!×8008$ ways
IF THE OBJECTS CAN BE REPEATED
If the object can be repeated, the $6$ spaces would give rise to other objects ( repeatition ) and the remaining space, More importantly let's not forget to include all the possible combination w.r.t repeating of the objects
Occurance of $1$ object and $5$ space
Occurance of $2$ object and $4$ space
Occurance of $3$ object and $3$ space
Occurance of $4$ object and $2$ space
Occurance of $5$ object and $1$ space
Occurance of $6$ objects ( no extra space )
So that the total number of ways to scatter the district object into the distinct boxes in repeatition, is the sum of the number of ways here
Therefore $10!×16C10×6×10 + 10!×16C10×6×(10+10C2) + 10!×16C10×6×(10+10C2+10C3) + 10!×16C10×6×(10+10C2+10C3+10C4) + 10!×16C10×6×(10+10C2+10C3+10C4+10C5) + 10!×16C10×6×(10+10C2+10C3+10C4+10C5+10C6)$
Which becomes
$10!×16C10×6×(10*6+10C2*5+10C3*4+10C4*3+10C5*2+10C6*1)$
$10!×16C10×6×2109$
The total number of ways in scattering the distinct objects is the sum of it ways in repeatition plus it ways in non-repeatition, the solution then becomes
$10!×16C10 + 10!×16C10×6×2109$
$10!×16C10×(1+12654) = 10!×16C10×12655$
$9,643,717,440,000$ ways
Have I done it right, or what's wrong with this approach, thanks for helping