How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?

256
128
240
224
192

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Number of subsets in a set of 8 elements is $2^8=256$ .

Subsets not containing a prime number contain only some of the four elements which are not prime numbers, namely $4, 6, 8, 9$ , or no elements at all.

There are $2^4=16$ subsets in a set of four elements, including the empty set.

The remaining subsets each contain at least one prime number and there are $256-16=\boxed{240}$ of them.