Explanation:

- We need to find values of \(p\) and \(q\) such that the total frequency is 120 and the mean is 50.

- Total frequency: \(17 + (p+q) + 32 + (p-3q) + 19 = 120\).

- Simplifies to \(2p - 2q + 68 = 120\).

- Further simplifies to \(2p - 2q = 52\).

- Dividing by 2: \(p - q = 26\).

- For the mean:

- Using midpoints of classes: \(10, 30, 50, 70, 90\).

- Weighted mean formula: \((17 \times 10) + ((p+q) \times 30) + (32 \times 50) + ((p-3q) \times 70) + (19 \times 90) = 50 \times 120\).

- Simplifies to: \(170 + 30p + 30q + 1600 + 70p - 210q + 1710 = 6000\).

- Further reduces to: \(100p - 180q = 2520\). Divide by 20: \(5p - 9q = 126\).

- Solving the equations:

- Equation 1: \(p - q = 26\)

- Equation 2: \(5p - 9q = 126\)

- Solving simultaneously gives \(p = 27\).

- Options:

- Option 1: 25

- Option 2: 26

- Option 3: 27

- Option 4: 28

- Correct Answer: Option 3: 27