How to find the number of distinct combinations of a non distinct set of elements?

Question

I'm trying to figure out a way to find the number of factors a number has based on its prime factorization without working out all the combinations. If the number breaks down into $n$ distinct prime factors I know it's max number of combinations is $2^n$, with $\frac{n!}{(n-r)!r!}$ being the number of combinations when selecting $r$ elements. However I don't know how to figure out the max number of combinations for a list of $n$ non-distinct primes, i.e. for $i$ from $1$ to $n$ the $i^{th}$ element repeats $i_{r}$ times. Is there any way to figure this number out without doing all the combinations and taking away the repeated ones?

Answer 1

If $n = \prod_{i=1}^m p_i ^{t_i} = p_1 ^ {t_1} p_2 ^{t_2} \cdots p_m ^{t_m}$, then the number of divisors of $n$ is given by $d(n) = \prod_{i=1}^m (t_i +1)$. Every divisor is of the form $n = \prod_{i=1}^m p_i ^{s_i}$ with some $s_i$ with $0 \leq s_i \leq t_i \, \forall i$. Working out the number of combinations gives the result. Did you mean this?

Answer 2

I agree with the answer by Zardo.

I suggest to stare at the formula for the number of divisors of n, d(n) given by Zardo.

Then try to find the number of divisors of a prime power, say $81 = 3^4$. You should see that there are 5: {1,3,9,27,81}. Then try to find the number of divisors of another prime power, say $125 = 5^3$.

You should see there are 4: {1,5,25,125}. Now you probably can see that there are 5*4 = 20 divisors of $10125 = 3^4*5^3$.

Note that the divisors of 10125 are the cross products of {1,3,9,27,81}X{1,5,25,125}.