Step 1 :Given that 13 students listened to rock and jazz, 20 students listened to rock and classical, 16 students listened to classical and jazz, and 10 students listened to all three styles.
Step 2 :To find the number of students who listened to exactly two styles of music, we need to subtract the number of students who listened to all three styles from the number of students who listened to each pair of styles.
Step 3 :Let's denote the number of students who listened to rock and jazz as \(rock\_and\_jazz\), the number of students who listened to rock and classical as \(rock\_and\_classical\), the number of students who listened to classical and jazz as \(classical\_and\_jazz\), and the number of students who listened to all three styles as \(all\_three\).
Step 4 :So, \(rock\_and\_jazz = 13\), \(rock\_and\_classical = 20\), \(classical\_and\_jazz = 16\), and \(all\_three = 10\).
Step 5 :Then, the number of students who listened to exactly two styles, denoted as \(exactly\_two\_styles\), can be calculated as \(rock\_and\_jazz + rock\_and\_classical + classical\_and\_jazz - 3 * all\_three\).
Step 6 :Substitute the given values into the equation, we get \(exactly\_two\_styles = 13 + 20 + 16 - 3 * 10 = 19\).
Step 7 :Final Answer: The number of students who listened to exactly two styles of music is \(\boxed{19}\).