PMS Provincial Management Services Solved Paper 2026

The correct answer is 720. The factorial notation (!) means multiplying all positive integers from the number down to 1. The expression 10! / 7! can be simplified by expanding the factorials and cancelling common terms.

Step-by-Step Solution

• Expand the Factorials:
  
  10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
  
  7! = 7 × 6 × 5 × 4 × 3 × 2 × 1
• Cancel the Common Factor 7!:
  
  10! / 7! = (10 × 9 × 8 × 7!) / 7!
  
  The 7! in the numerator and denominator cancel out completely.
• Multiply the Remaining Numbers:
  
  10 × 9 × 8 = 720
• Verification:
  
  10! = 3,628,800 and 7! = 5,040.
  
  3,628,800 ÷ 5,040 = 720. Confirmed.

Thus, 10! / 7! equals 720.

The correct answer is 300. We are told that 45% of the participants are boys, and the number of boys is 135. Using this information, we can find the total number of students by setting up a simple percentage equation.

Step-by-Step Solution

• Understand the Relationship:
  
  45% of the total students = Number of boys
  
  Let the total number of students be T.
• Write the Equation:
  
  45% × T = 135
  
  (45/100) × T = 135
• Solve for T:
  
  Multiply both sides by 100 to eliminate the denominator:
  
  45 × T = 135 × 100
  
  45T = 13,500
  
  Divide both sides by 45:
  
  T = 13,500 ÷ 45
  
  T = 300
• Verification:
  
  45% of 300 = (45/100) × 300 = 0.45 × 300 = 135. The answer matches the given number of boys perfectly.

Hence, the total number of students participating is 300.

The correct answer is -1. This is a basic linear equation. We will isolate the variable x step by step using algebraic operations.

Step-by-Step Solution

• Write the Given Equation:
  
  3(x - 1) = -6
• Distribute the 3 on the Left Side (or Divide First):
  
  Method 1 – Divide first:
  
  Divide both sides by 3 to remove the coefficient:
  
  (x - 1) = -6 ÷ 3
  
  x - 1 = -2
• Isolate x:
  
  Add 1 to both sides:
  
  x - 1 + 1 = -2 + 1
  
  x = -1
• Verification:
  
  Substitute x = -1 back into the original equation:
  
  3(-1 - 1) = 3(-2) = -6. The left side equals the right side, confirming the solution.

Therefore, the value of x is -1.

The correct answer is 25%. Percentage increase is calculated by finding the absolute increase (new value minus original value), dividing it by the original value, and then multiplying by 100.

Step-by-Step Solution

• Identify Original and New Lengths:
  
  Original length = 72 cm
  
  New length = 90 cm
• Calculate the Absolute Increase:
  
  Increase = New length - Original length
  
  Increase = 90 - 72 = 18 cm
• Compute the Percentage Increase:
  
  Percentage Increase = (Increase ÷ Original length) × 100
  
  = (18 ÷ 72) × 100
  
  = 0.25 × 100
  
  = 25%
• Verification:
  
  25% of 72 cm = (25/100) × 72 = 18 cm.
  
  72 cm + 18 cm = 90 cm, which matches the new length exactly.

Thus, the increasing percentage is 25%.

The correct answer is 50p² + 72q². This algebraic simplification uses the identities for the square of a binomial: (a - b)² = a² - 2ab + b² and (a + b)² = a² + 2ab + b². When added, the middle terms cancel beautifully.

Step-by-Step Solution

• Expand the First Square (5p - 6q)²:
  
  Using (a - b)² = a² - 2ab + b², with a = 5p and b = 6q:
  
  (5p)² - 2×(5p)×(6q) + (6q)²
  
  = 25p² - 60pq + 36q²
• Expand the Second Square (5p + 6q)²:
  
  Using (a + b)² = a² + 2ab + b²:
  
  (5p)² + 2×(5p)×(6q) + (6q)²
  
  = 25p² + 60pq + 36q²
• Add the Two Expanded Expressions:
  
  (25p² - 60pq + 36q²) + (25p² + 60pq + 36q²)
  
  Combine like terms:
  
  25p² + 25p² = 50p²
  
  -60pq + 60pq = 0 (the middle terms cancel out)
  
  36q² + 36q² = 72q²
  
  Result: 50p² + 72q²
• Verification:
  
  Let p = 1, q = 1. Original expression: (5-6)² + (5+6)² = (-1)² + (11)² = 1 + 121 = 122.
  
  Our result: 50(1)² + 72(1)² = 50 + 72 = 122. Matches.

Thus, the simplified expression is 50p² + 72q².

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The correct answer is 28. To find the unknown number, we translate the percentage statement into an algebraic equation and solve for the variable.

Step-by-Step Solution

• Step 1 – Understand the meaning of 175%:
  
  Percent means \"\"per hundred\"\", so 175% = 175/100 = 1.75 in decimal form.
• Step 2 – Set up the equation:
  
  Let the required number be x.
  
  \"\"175% of x is 49\"\" translates to:
  
  1.75 × x = 49
• Step 3 – Isolate x:
  
  Divide both sides of the equation by 1.75:
  
  x = 49 ÷ 1.75
• Step 4 – Simplify the division:
  
  To avoid decimals, multiply the numerator and denominator by 100:
  
  x = (49 × 100) ÷ (1.75 × 100) = 4900 ÷ 175
  
  Now divide: 4900 ÷ 175 = 28.
• Step 5 – Verification:
  
  Find 175% of 28:
  
  (175/100) × 28 = 1.75 × 28 = 49.
  
  The result matches the given value, confirming the solution.

Hence, the number is 28.

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The correct answer is A ∩ B = φ. This notation states that the intersection of events A and B is the empty set, which perfectly describes mutual exclusivity.

Step-by-Step Explanation

• Step 1 – Define mutually exclusive events:
  
  Two events are mutually exclusive if they cannot occur simultaneously. For example, when rolling a die, getting a "2" and getting a "5" on the same single roll are mutually exclusive; both outcomes cannot happen at once.
• Step 2 – Understand the set notation:
  
  In probability, events are represented as sets of outcomes.
  
  A ∩ B (A intersection B) is the set of outcomes that belong to both A and B.
  
  φ (the Greek letter phi) denotes the empty set, meaning there are no elements.
• Step 3 – Apply the definition:
  
  For mutually exclusive events, there is no common outcome. Therefore, the set containing outcomes common to both A and B is empty.
  
  This is written as: A ∩ B = φ
• Step 4 – Eliminate incorrect options:
  
  A ∩ B = S: This would mean the intersection is the entire sample space, implying A and B always happen together (the exact opposite of mutual exclusivity).
  
  A ∪ B = φ: This would mean that neither A nor B ever happens, which is impossible for valid events.
  
  A = B: If A and B are the same event, they always occur together, again contradicting mutual exclusivity.
• Step 5 – Probability consequence:
  
  For mutually exclusive events, P(A ∩ B) = 0, and the addition rule simplifies to P(A ∪ B) = P(A) + P(B).

Therefore, the correct relation for mutually exclusive events A and B is A ∩ B = φ.