A 3-digit number is the product of 4 prime numbers. Given that the 3 digits of the number are all prime & different and that the sum of ...its...

The prime digits are 2,3,5 and
7.

So we can form only 4! numbers which are different 3
digit prime numbers. So our choice number is from of these 24 numbers. In fact these
could be physically tried. But a short cut is always better. So we proceed as
shown below:

 Since the factors of the number are 4 prime
numbers, which add up to 30, we should have the factors 2,3,5,7,11,13,17,19,23 and
29.

Examining the different primes from 2 to 29 , no 4
different primes add up to 30.

However , the conditions
given say only different prime 3 digits. But the condition does not say the 4
factors should  be different primes.

If  the  3 digit prime
number allows for 4 prime factors , not neccesarily different, then 2*2*7*19 =
532 is a
solution.

Tally:

Number
of factors are four .

Factors are: 2, 2, 7 and
19.

All are prime factors.

Sum
of the factors: 2+2+7+19 = 30.

The digits of 532 are  5, 3
and 2 are three  distinct primes.

Hope this
helps.