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Probability & Permutation Basics

Quantitative Aptitude / DI · Advanced Quant / Data

Useful for: IBPS Banking

Overview

Permutations count arrangements, combinations count selections, and probability divides favourable by total outcomes.

Key points

nPr = n!/(n − r)!; nCr = n!/[r!(n − r)!].
Arrangements of n letters with repeats: n!/(p! q! …).
Probability = favourable ÷ total equally likely outcomes.
Circular arrangements of n objects: (n − 1)!.
nCr = nC(n − r).

Explanation

"Committee/selection" → combination; "arrangement/order" → permutation.
Treat items that must stay together as one unit.
For "at least one", use 1 − P(none).
Cards: 52 cards, 4 kings, 26 red.

Step by step

Translate the data or statement into numbers.
Compute step by step with the right formula.
Compare with options; check reasonableness.

Examples

Arrangements of the letters of BANK = 4! = 24.
Choosing 2 people from 5 = 5C2 = 10.
Arrangements of APPLE = 5!/2! = 60.

Traps and advanced points

Trap: Using permutation for selections.
Trap: Forgetting to divide for repeated letters.
Trap: Linear formula for circular arrangements.

Quick revision

nPr order; nCr selection.
Repeats → divide.
Circle → (n − 1)!.

Practice MCQs

Group A has 15 observations with mean 103; Group B has 19 observations with mean 201. Find the combined mean.
1. 157.76
2. 152
3. 158.76
4. 156.76
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Answer: (a) 157.76. Combined mean = total of both groups ÷ combined count.
Group A has 16 observations with mean 80; Group B has 18 observations with mean 155. Find the combined mean.
1. 120.71
2. 117.5
3. 119.71
4. 118.71
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Answer: (c) 119.71. Combined mean = total of both groups ÷ combined count.
Group A has 18 observations with mean 74; Group B has 15 observations with mean 143. Find the combined mean.
1. 104.36
2. 108.5
3. 105.36
4. 106.36
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Answer: (c) 105.36. Combined mean = total of both groups ÷ combined count.
A bag contains 10 red, 69 blue and 119 green balls. One ball is drawn at random. Probability of drawing a red ball?
1. 69/198
2. 119/198
3. 10/188
4. 10/198
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Answer: (d) 10/198. Probability = favourable outcomes ÷ total equally likely outcomes.
A bag contains 12 red, 71 blue and 123 green balls. One ball is drawn at random. Probability of drawing a red ball?
1. 71/206
2. 123/206
3. 12/206
4. 12/194
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Answer: (c) 12/206. Probability = favourable outcomes ÷ total equally likely outcomes.
Arrangements of the letters of "BOOK" —
1. 12
2. 6
3. 4
4. 24
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Answer: (a) 12. 4!/2! = 12.
Arrangements of the letters of "LEVEL" —
1. 20
2. 120
3. 60
4. 30
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Answer: (d) 30. 5!/(2! 2!) = 30.
5 people around a round table can sit in —
1. 60 ways
2. 5 ways
3. 120 ways
4. 24 ways
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Answer: (d) 24 ways. (5 − 1)! = 24.
Probability of a sum of 7 with two dice —
1. 1/6
2. 1/12
3. 1/36
4. 7/36
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Answer: (a) 1/6. 6/36.
Probability of drawing a king from a pack of 52 cards —
1. 1/13
2. 1/52
3. 4/13
4. 1/4
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Answer: (a) 1/13. 4/52.