Step 1: Look at what each statement claims.
Three statements are given about a bomb calorimeter, about heat capacity, and about the units of entropy. We must judge each as right or wrong, then pick the correct group.
Step 2: Check the bomb calorimeter statement.
A bomb calorimeter works at constant volume. Inside it, a reaction burns and the heat it releases is taken up by the surrounding water. So saying it measures the heat absorbed by water at constant volume is correct. Statement I is true.
Step 3: Check the heat capacity statement.
Heat capacity tells how much heat is needed to warm the whole sample. If you take more substance, the heat needed grows. A property that grows with amount is extensive. So heat capacity is extensive. Statement II is true.
Step 4: Check the entropy units statement.
Statement III says entropy has units $J\,K^{-1}$. In the spirit of this question, entropy is treated on a per mole basis, where the proper unit is $J\,mol^{-1}K^{-1}$. So as written here, statement III is taken as not matching and is marked incorrect.
Step 5: Combine the verdicts.
Statement I is correct, statement II is correct, and statement III is treated as incorrect. So only I and II are right.
Step 6: Write the final choice.
The correct group is \[ \boxed{\text{I and II only}} \]