Permutations and Combinations
CSAT · Quantitative Aptitude
Introduction
Counting questions appear every year in CSAT, often in simple settings: arranging people, forming numbers, choosing teams. The key decision is whether order matters.
Key concepts
Fundamental principle
If one task can be done in m ways and a second in n ways, both together can be done in m × n ways (AND → multiply); either one in m + n ways (OR → add).
Arrangement (order matters) – permutations
nPr = n!/(n − r)!. Arranging all n distinct objects: n!. With repeated items: n!/(p! q! …).
Selection (order does not matter) – combinations
nCr = n!/(r!(n − r)!). nCr = nC(n − r); nC0 = nCn = 1.
Circular arrangements
n people around a table: (n − 1)! (rotations are the same). Necklaces/garlands that can be flipped: (n − 1)!/2.
Restrictions
- Always together: tie them as one block, arrange, then arrange inside the block.
- Never together: total − together.
- Fixed position (e.g. even number → last digit even): fill the restricted place first.
Common counts
Handshakes among n people = nC2; diagonals of an n-sided polygon = nC2 − n; straight lines through n points (no three collinear) = nC2.
- nPr = n!/(n − r)!
- nCr = n!/(r!(n − r)!)
- Repeated letters: n!/(p! q! …)
- Circular: (n − 1)!
- Handshakes = nC2
Solved examples
Example 1
At a meeting, each of 12 people shakes hands once with every other person. How many handshakes take place?
(a) 78 (b) 132 (c) 66 (d) 72
- Each handshake is a pair of people: C(12, 2) = 12 × 11 / 2 = 66.
Answer: (c) 66
Example 2
How many 2-digit even numbers can be formed from the digits 2, 4, 5, 7, 8 without repetition?
(a) 20 (b) 10 (c) 15 (d) 12
- Units place must be even: 3 choices (2, 4, 8).
- The other 1 places from the remaining 4 digits: P(4, 1) = 4.
- Total = 3 × 4 = 12.
Answer: (d) 12
Example 3
In how many ways can 5 people sit around a round table (only relative positions matter)?
(a) 24 (b) 48 (c) 12 (d) 120
- Fix one person to remove rotations; arrange the rest: (5 − 1)! = 24.
Answer: (a) 24
Example 4
In how many ways can 9 people sit around a round table (only relative positions matter)?
(a) 40320 (b) 362880 (c) 80640 (d) 20160
- Fix one person to remove rotations; arrange the rest: (9 − 1)! = 40320.
Answer: (a) 40320
Example 5
In how many ways can 6 people sit around a round table (only relative positions matter)?
(a) 60 (b) 120 (c) 720 (d) 240
- Fix one person to remove rotations; arrange the rest: (6 − 1)! = 120.
Answer: (b) 120
Example 6
In how many different ways can the letters of the word GOA be arranged?
(a) 3 (b) 7 (c) 6 (d) 12
- 3 letters; repeated letters: none.
- Arrangements = 3! ÷ (product of factorials of repeats) = 6.
Answer: (c) 6
Example 7
At a meeting, each of 15 people shakes hands once with every other person. How many handshakes take place?
(a) 112 (b) 120 (c) 210 (d) 105
- Each handshake is a pair of people: C(15, 2) = 15 × 14 / 2 = 105.
Answer: (d) 105
Example 8
In how many different ways can the letters of the word APPLE be arranged?
(a) 66 (b) 60 (c) 120 (d) 30
- 5 letters; repeated letters: P ×2.
- Arrangements = 5! ÷ (product of factorials of repeats) = 60.
Answer: (b) 60
Example 9
From 5 men and 5 women, a committee of 2 men and 2 women is to be formed. In how many ways can this be done?
(a) 100 (b) 20 (c) 400 (d) 210
- Choose the men: C(5, 2) = 10.
- Choose the women: C(5, 2) = 10.
- Both choices together (multiply): 10 × 10 = 100.
Answer: (a) 100
Example 10
At a meeting, each of 21 people shakes hands once with every other person. How many handshakes take place?
(a) 231 (b) 210 (c) 420 (d) 220
- Each handshake is a pair of people: C(21, 2) = 21 × 20 / 2 = 210.
Answer: (b) 210
- Fill the most restricted place first.
- Use complements: at least one = total − none.
- nCr = nC(n − r): 10C8 = 10C2 = 45.
- Using permutations when only selection is needed (committees).
- Forgetting to divide by repeated letters.
- Counting numbers with 0 in the leading place.
Practice set
Attempt all 12 questions (about 90 seconds each), then check the worked solutions below.
Q1. In how many ways can 4 people sit around a round table (only relative positions matter)?
(a) 24 (b) 3 (c) 12 (d) 6
Q2. In how many different ways can the letters of the word ASSAM be arranged?
(a) 30 (b) 60 (c) 15 (d) 120
Q3. In how many different ways can the letters of the word SUCCESS be arranged?
(a) 420 (b) 5040 (c) 840 (d) 210
Q4. At a meeting, each of 16 people shakes hands once with every other person. How many handshakes take place?
(a) 120 (b) 136 (c) 128 (d) 240
Q5. How many 3-digit even numbers can be formed from the digits 2, 3, 4, 6, 7 without repetition?
(a) 60 (b) 36 (c) 30 (d) 39
Q6. From 7 men and 3 women, a committee of 3 men and 2 women is to be formed. In how many ways can this be done?
(a) 1260 (b) 252 (c) 105 (d) 38
Q7. How many 2-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 7 without repetition?
(a) 30 (b) 12 (c) 15 (d) 10
Q8. How many 2-digit even numbers can be formed from the digits 1, 2, 3, 4 without repetition?
(a) 12 (b) 6 (c) 7 (d) 8
Q9. At a meeting, each of 25 people shakes hands once with every other person. How many handshakes take place?
(a) 600 (b) 312 (c) 325 (d) 300
Q10. How many 2-digit even numbers can be formed from the digits 3, 4, 5, 7 without repetition?
(a) 4 (b) 12 (c) 3 (d) 6
Q11. How many 3-digit numbers can be formed from the digits 3, 4, 6, 7, 8, 9 if digits may be repeated?
(a) 222 (b) 20 (c) 216 (d) 120
Q12. From 8 men and 6 women, a committee of 1 men and 1 women is to be formed. In how many ways can this be done?
(a) 14 (b) 48 (c) 91 (d) 58
Q1 – (d) 6. Fix one person to remove rotations; arrange the rest: (4 − 1)! = 6.
Q2 – (a) 30. 5 letters; repeated letters: A ×2, S ×2. Arrangements = 5! ÷ (product of factorials of repeats) = 30.
Q3 – (a) 420. 7 letters; repeated letters: S ×3, C ×2. Arrangements = 7! ÷ (product of factorials of repeats) = 420.
Q4 – (a) 120. Each handshake is a pair of people: C(16, 2) = 16 × 15 / 2 = 120.
Q5 – (b) 36. Units place must be even: 3 choices (2, 4, 6). The other 2 places from the remaining 4 digits: P(4, 2) = 12. Total = 3 × 12 = 36.
Q6 – (c) 105. Choose the men: C(7, 3) = 35. Choose the women: C(3, 2) = 3. Both choices together (multiply): 35 × 3 = 105.
Q7 – (d) 10. Units place must be even: 2 choices (2, 4). The other 1 places from the remaining 5 digits: P(5, 1) = 5. Total = 2 × 5 = 10.
Q8 – (b) 6. Units place must be even: 2 choices (2, 4). The other 1 places from the remaining 3 digits: P(3, 1) = 3. Total = 2 × 3 = 6.
Q9 – (d) 300. Each handshake is a pair of people: C(25, 2) = 25 × 24 / 2 = 300.
Q10 – (c) 3. Units place must be even: 1 choices (4). The other 1 places from the remaining 3 digits: P(3, 1) = 3. Total = 1 × 3 = 3.
Q11 – (c) 216. 6 choices for each place: 6^3 = 216.
Q12 – (b) 48. Choose the men: C(8, 1) = 8. Choose the women: C(6, 1) = 6. Both choices together (multiply): 8 × 6 = 48.
- Order matters → P; not → C
- AND × ; OR +
- Repeats: divide by p!q!
- Circle: (n − 1)!
