
number of factorisations
How many different nontrivial factorisations does
3 × 5 × 7 × 11 × 13 × 17 × 19 × 23 × 29 × 31 = 100280245065.
have?
Since there are 10 primes, i thought the answer would be
sum_(i=0)^n {10 \choose i}  1 = 2^10  1
but i am a little bit unsure if i am not counting anything double.

Try with $\displaystyle 3 \times 5 \times 7$ and $\displaystyle 3 \times 5 \times 7 \times 11$ (do it by hand, boring but useful), look for a pattern, check it with 5 prime factors, write it in maths, then apply it on your 10 prime factors :)

I was wondering if the following is a correct argument:
Let n be the number as before and consider the set A:={3,5,7,11,13,17,19,23,29,31}. The product of the elements of each subset B of this set give a different divisor of the given number. However, the factorisation obtained by diving b by the product of the elements in some subset B, is the same as the factorisation obtained by diving n by the product of the elements in A\B.
The total number of subsets of A is equal to the number of elements in P(A)=power set of A. However, this includes the empty set and A itself, which yield a trivial factorisation. Thus the total number of nontrivial factorisations is 1/2*(2^(10)2)=2^91.

It seems to work with $\displaystyle n = 3$, because we have manually $\displaystyle 3$ arrangements, and with your formula : $\displaystyle 2^{n  1}  1 = 2^2  1 = 4  1 = 3$