This would be the generic strategy:

Step 1: Find out the number of products you get.
5 distinct numbers in set A and 4 distinct in set B so number of products = 5*4 = 20

Step 2: Remove the products that appear more than once.
Notice that 5, 7, 11 and 13 are primes and none of their multiples are in either set. So ignore them. We just need to focus on 2 and 3 of set A and 2, 4 and 6 of set B.

2, 3
2, 4, 6
The only product repeated when you take a number from each set is 12. (3*4 and 2*6) Rest all are distinct.

Answer = 20 - 1 = 19

Note here that the second step will involve manual calculation since it will depend on the specific numbers you have in the two sets.
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It can be easily done by making pairs, but can somebody tell me how to do it using combinations or some other short-cut ?

Karishma,

I don't understand this part of your explanation:
"Notice that 5, 7, 11 and 13 are primes and none of their multiples are in either set. So ignore them. We just need to focus on 2 and 3 of set A and 2, 4 and 6 of set B."

Why do we have to do that? Thanks!

This is part of your step 2: Remove the products that appear more than once.
The logic here is that a product involving 5/7/11/13 will not appear more than once. So we ignore these numbers.

Say we select 5 from A. Now, when we select any number from set B, we get a distinct product i.e. we get 4 distinct products (5*2, 5*4, 5*6, 5*13)
Now think, can you select a number other than 5 from set A and some number from set B to make one of these 4 products? i.e. Without selecting 5 from set A, can you make a product of 10 or 20 or 30 or 65? No, because to make 10/20/30/65, you need a 5 but you have no other 5 or multiple of 5.
Same is the case with 7, 11 and 13 (primes that appear only once in one set). So the products made by these prime numbers will not appear more than once.

You don't really need to think all this during your test. Lots of practice and thorough analysis will make these things intuitive.
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By looking at Set A, we can see that it's all primes. Thus, we should immediately break down the elements in Set B to their prime factors. That gives :

Set A = {2,3,5,7,11}

Set B = {2, 2x2, 3x2, 13}

Apart from 2x3x2 (taking 2 from set A) which is the same as 3x2x2(taking 3 from set A); there is nothing which can be repeated. Thus, the total unique product = 20-1 = 19.
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Total values will be \(5C_1 * 4C_1\)

But there will one repetitive value as 12

20 - 1

19
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