Civil Service Exam (Subprofessional) Numerical Ability — Permutation & CombinationFlash Cards

What is the key difference between permutation and combination?

Permutation considers the ORDER of arrangement (arrangement matters), while combination does NOT consider order (selection only matters). Example: ABC and BAC are different in permutation but same in combination.

What is the formula for permutation nPr?

nPr = n! / (n-r)! Where: - n = total number of objects - r = number of objects selected - ! = factorial Example: 5P3 = 5! / (5-3)! = 5! / 2! = 60

What is the formula for combination nCr?

nCr = n! / (r!(n-r)!) Where: - n = total number of objects - r = number of objects selected - ! = factorial Example: 5C3 = 5! / (3!×2!) = 120 / (6×2) = 10

What is factorial and how do you calculate it?

Factorial (n!) is the product of all positive integers from 1 to n. Examples: - 3! = 3 × 2 × 1 = 6 - 5! = 5 × 4 × 3 × 2 × 1 = 120 - 0! = 1 (by definition) Used in both permutation and combination formulas.

When should you use PERMUTATION in a problem?

Use permutation when: - ORDER matters - Arrangement is important - Keywords: arrange, order, sequence, ranking Example: 'In how many ways can 5 students line up?' - Order matters, so use permutation.

When should you use COMBINATION in a problem?

Use combination when: - ORDER does NOT matter - Only selection is important - Keywords: choose, select, pick, group Example: 'Choose 3 students from 10 for a committee' - Order doesn't matter, so use combination.

How many ways can a director choose 3 of their 10 talented artists to take part in an art class?

This is a COMBINATION problem (choosing/selecting). Given: n = 10, r = 3 10C3 = 10! / (3!(10-3)!) = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120 ways

In how many ways can the letters of 'ALPHA' be arranged?

This is a PERMUTATION problem (arrangement). Since all letters are different: 5P5 = 5! = 5 × 4 × 3 × 2 × 1 = 120 ways Alternatively, since we're using all letters, it's simply 5! = 120

What are consecutive numbers and give examples?

Consecutive numbers follow each other in order without skipping. Examples: - Consecutive integers: 1, 2, 3, 4, 5 - Consecutive even: 2, 4, 6, 8, 10 - Consecutive odd: 1, 3, 5, 7, 9 Pattern: If first number is n, then n, n+1, n+2, n+3... For even/odd: n, n+2, n+4, n+6...

What is the difference between consecutive even and consecutive odd numbers?

Both consecutive even and consecutive odd numbers have a difference of 2 between adjacent numbers. Consecutive even: 22, 24 → difference = 2 Consecutive odd: 11, 13 → difference = 2 For any two consecutive even numbers: difference = 2 For any two consecutive odd numbers: difference = 2

If the sum of two consecutive numbers is 9, find the smaller number.

Let x = smaller number, x+1 = larger number Equation: x + (x+1) = 9 2x + 1 = 9 2x = 8 x = 4 Therefore, the smaller number is 4. Check: 4 + 5 = 9 ✓

A security system uses a 4-letter password with no repeated letters. How many possible passwords are there?

This is a PERMUTATION problem (order matters for passwords). Total letters in alphabet = 26 Selecting 4 letters without repetition: 26P4 = 26! / (26-4)! = 26! / 22! = 26 × 25 × 24 × 23 = 358,800 possible passwords

Compare nPr and nCr. Which is larger and why?

nPr is ALWAYS larger than or equal to nCr for the same values of n and r. Reason: nPr = n!/(n-r)! while nCr = n!/(r!(n-r)!) Since nCr has an additional r! in the denominator, it's smaller. Relationship: nPr = nCr × r! Example: 5P3 = 60, 5C3 = 10, and 60 = 10 × 3!

In a national election, voters select 12 senators from 15 candidates. How many ways can this be done?

This is a COMBINATION problem (selection, order doesn't matter). Given: n = 15, r = 12 15C12 = 15! / (12!(15-12)!) = 15! / (12! × 3!) = (15 × 14 × 13) / (3 × 2 × 1) = 2730 / 6 = 455 ways

What keywords indicate a COMBINATION problem?

Keywords for COMBINATION: - Choose - Select - Pick - Group - Committee - Team formation Remember: These suggest that ORDER DOESN'T MATTER, only the selection itself matters.

What keywords indicate a PERMUTATION problem?

Keywords for PERMUTATION: - Arrange - Order - Sequence - Ranking - Position - Line up - Password/codes Remember: These suggest that ORDER MATTERS in the arrangement.

Sum of three consecutive numbers is 63. Find the highest number.

Let x = highest number Then: (x-2) + (x-1) + x = 63 Solving: 3x - 3 = 63 3x = 66 x = 22 Highest number = 22 Check: 20 + 21 + 22 = 63 ✓

Five candidates run for president, vice president, and secretary. How many different combinations can fill these three positions?

This is a PERMUTATION problem because positions are different (order matters). We're selecting 3 positions from 5 candidates: 5P3 = 5! / (5-3)! = 5! / 2! = (5 × 4 × 3 × 2 × 1) / (2 × 1) = 120 / 2 = 60 different combinations

What is the sum property of two consecutive numbers?

The sum of two consecutive numbers is ALWAYS an odd number. Examples: - 1 + 2 = 3 (odd) - 5 + 6 = 11 (odd) - 20 + 21 = 41 (odd) This happens because: even + odd = odd, and consecutive numbers are always one even and one odd.

A company has 5 vacancies for junior auditor positions from 7 candidates. How many groups of five can be chosen?

This is a COMBINATION problem (choosing groups, order doesn't matter). Given: n = 7, r = 5 7C5 = 7! / (5!(7-5)!) = 7! / (5! × 2!) = (7 × 6) / (2 × 1) = 42 / 2 = 21 different groups