Syd chooses two different primes, both of which are greater than $10,$ and multiplies them. The resulting product is less than $

The answer is 10. Between 10 and 35, there are 7 prime numbers: 11, 13, 17, 19, 23, 29, and 31. Multiplying 11 by any of the others yields a product smaller than 350, resulting in 6 products. The product of 13 with anything below 26 will also be less than 350, adding 3 more products. Similarly, the product of 17 with anything below 20 yields 1 additional product. Therefore, the total count of different products under 350 amounts to 10.