Total number of combinations.

Question

Total number of combination of $a,b,c$ and $d$ where elements can repeat up to maximum $m,n,o,p$ times respectively.(Note: order does not matter , i.e., ${a,b}$ and ${b,a}$ will be count as one only).
eg: {$a,b,c$} where $'a'$ can come upto 2 times i.e, 'a' can come in combination from 0 to 2 times, similarly $'b'$ can come at max 2 times & $'c'$ can come at max 1 times.
so different combinations can be {$a$},{$b$},{$c$},{$a,a,b,c$},....,{$a,a,b,b,c$}.
remember {$a,a,a$} is not a solution because a occurs more than 2 times in combination.

gandalf61 · Answer

Term $a$ can occur from $0$ to $m$ times, so there are $m+1$ possibilities for the number of $a$ terms.
Similarly $b$ can occur from $0$ to $n$ times, so there are $n+1$ possibilities for the number of $b$ terms etc.
Once you know the number of $a, b, c$ and $d$ terms respectively then you have completely defined the combination, since order does not matter.
This is essentially the same problem as finding the number of divisors of $a^mb^nc^od^p$ where $a, b, c$ and $d$ are distinct primes.

Total number of combinations.

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