There are 7 primes between 10 and 35: 11, 13, 17, 19, 23, 29, 31.

The product of 11 and any of the others will be less than 350, 6 products.

The product of 13 and any below 26 will be less than 350, 3 more products.

The product of 17 and any below 20 will be less than 350, 1 more product.

There are a total of 10 different products below 350 possible