The number of people who like only basketball is: 120 - 50 = 70.

The number of people who like only football is 120 - 40 = 80.

The number of people who like only football is 120 - 50 = 70.

The number of people who like at least one sport is: 120 + 80 - 50 = 150. Therefore, those who like neither is: 200 - 150 = 50.

The number of people who like only music is 80 - 50 = 30.

The number of people who like only rock music is 150 - 50 = 100.

Students who like only vanilla = Total vanilla - Both = 20 - 10 = 10.

The reciprocals are 1 and 2, which are in arithmetic progression with a common difference of 1/2.

The first term is 1, the second term is 1/2, and the third term can be calculated as 1/(1 + 1/2) = 2/3. The sum is 1 + 1/2 + 2/3 = 2.

The reciprocals are 1 and 2, which are in arithmetic progression. The third term's reciprocal is 3, so the third term is 1/3.

In a harmonic progression, the reciprocals of the terms form an arithmetic progression. The reciprocals of 2 and 3 are 1/2 and 1/3. The common difference is 1/3 - 1/2 = -1/6. The third term's reciprocal will be 1/3 - 1/6 = 1/6, so the third term is 6.

In a harmonic progression, the reciprocals of the terms form an arithmetic progression. The reciprocals of 2 and 4 are 1/2 and 1/4. The common difference is -1/4. Therefore, the third term's reciprocal is 1/4 - 1/4 = 0, which means the third term is 1.

In a harmonic progression, the reciprocals of the terms form an arithmetic progression. The reciprocals of the first two terms are 1/2 and 3/4. The common difference is 1/4, so the reciprocal of the third term is 1/2 + 1/4 = 3/4. Therefore, the third term is 1/(3/4) = 4/3.

The reciprocals of the terms are 1/3 and 1/6. The common difference is (1/6 - 1/3) = -1/6. The third term's reciprocal will be 1/6 - 1/6 = 0, which means the third term is 1/12, thus the answer is 12.

The reciprocals of 3 and 6 are 1/3 and 1/6, which are in arithmetic progression.

The reciprocals of the first two terms are 1/3 and 1/6. The common difference is 1/6 - 1/3 = -1/6, which is incorrect. The correct common difference is 1/3 - 1/6 = 1/6.

The reciprocals of the terms are 1/4 and 1/2. The common difference is 1/2 - 1/4 = 1/4.

The reciprocals are 1/4 and 1/8. The common difference is 1/8 - 1/4 = -1/8, which means the common difference of the corresponding arithmetic progression is 1/8.

The reciprocals are 1/4 and 1/8. The common difference is 1/8 - 1/4 = -1/8. The third term's reciprocal will be 1/8 - 1/8 = 0, hence the third term is 16.

The first term is 5, and the second term in the harmonic progression corresponds to the reciprocal of the second term in the arithmetic progression, which is 5 + 2 = 7. Thus, the second term is 1/7.

The reciprocals are 1/5 and 1/10. The common difference is 1/10 - 1/5 = -1/10, which is the difference in the arithmetic progression.

The reciprocals are 1/5 and 1/10. The common difference is -1/10. The reciprocal of the third term is 1/10 - 1/10 = 1/15, so the third term is 15.

The nth term of a harmonic progression can be expressed as 1/(1/a + (n-1)d) where d is the common difference of the corresponding arithmetic progression.

'1A' in hexadecimal is 1*16 + 10 = 26 in decimal.

In hexadecimal, 'A' represents 10 and '3' represents 3. Therefore, A3 = (10 * 16^1) + (3 * 16^0) = 160 + 3 = 163.

The number of books that are only fiction is calculated by subtracting the number of books that are both from the total fiction books: 50 - 10 = 40.

The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Here, m = 3 and b = -2, so the equation is y = 3x - 2.

The slope indicates the steepness of the line, representing the ratio of the rise over the run.

In logic, if 'P implies Q' is true, it means that whenever P is true, Q must also be true.

A premise is a statement or proposition that provides the basis for a conclusion in a logical argument.