GMAT Prep Book 3: Permutation Combination

Permutation combination is considered a hard math topic by many GMAT test takers. It is believed that questions from this topic appear when you score in the higher percentile in the GMAT Quant section. Rest assured that GMAT tests questions from this topic that range from easy to medium level of difficulty. Therefore, a thorough understanding of the basics should set you on the path to cracking questions in the GMAT from this topic.

Syllabus Covered in Wizako's GMAT Book | Combinatorics

Wizako's Math Lesson Book in this chapter covers concepts right from absolute basics. Most formulae used are derived after explaining the basis with simple examples and in many cases by listing down the number of possibilities. The chapter includes the following concepts:

1. Independent events, product rule, sampling with and without replacement, sampling with and without ordering (arrangement).
2. Introduction to permutation, combination. Difference between permutation and combination. npr and ncr Formulae.
3. Examples of sampling with replacements, r-sequence and r-multisets.
4. Solved examples involving permutation and combination concepts in listing numbers
5. Solved examples involving re-arranging letters of words and their ranks
6. Concepts and solved examples on tossing of coins
7. Concepts and solved examples on rolling of a die and multiple dice
8. Solved examples on drawing one or more cards from a pack of cards
9. Typical permutation problems such as arranging boys and girls in a line etc.,
10. Typical combination problems such as questions on making musical albums, chess boards etc.,
11. Concept of circular permutation
12. 5 illustrative examples to explain concepts; 56 solved examples to acquaint you with as many different questions as possible
13. 24 exercise problems with answer key and explanatory answers to provide you with practice
14. A multiple choice question test with 60 GMAT level questions in the work book. An answer key and explanatory answer for all questions have been provided.

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Here is a typical solved example in Wizako's GMAT Book from this chapter

Sample Question

Each of the 11 letters A, H, I, M, O, T, U, V, W, X and Z appears same when looked at in a mirror. They are called symmetric letters. Other letters in the alphabet are asymmetric letters.
How many three letter computer passwords can be formed (no repetition allowed) with at least one symmetric letter?

Explanatory Answer

There are three possible cases that will satisfy the condition of forming three letter passwords with at least 1 symmteric letter.

Case 1: 1 symmetric and 2 asymmetric
Case 2: 2 symmetric and 1 asymmetric
Case 3: all 3 symmetric

Number of selections for case 1

1 symmetric letter can be selected from 11 in ¹¹C₁ ways.
2 asymmetric letters can be selected from the remaining 15 letters in ¹⁵C₂ ways.
Number of ways of selecting 1 symmetric and 2 asymmetric = ¹¹C₁ × ¹⁵C₂

Number of selections for case 2

2 symmetric letters can be selected from 11 in ¹¹C₂ ways.
1 asymmetric letter can be selected from the remaining 15 letters in ¹⁵C₁ ways.
Number of ways of selecting 2 symmetric and 1 asymmetric = ¹¹C₂ × ¹⁵C₁

Number of selections for case 3

3 symmetric letters can be selected from 11 in ¹¹C₃ ways.

Do not forget the rearrangements possible

The 3 distinct letters chosen can be re arranged in 3! ways

Total number of passwords that can be formed = {(¹¹C₁ × ¹⁵C₂) + (¹¹C₂ × ¹⁵C₁) + ¹¹C₃} × 3!

= {11 × \\frac{\text{15 × 14}}{\text{1 × 2}}) + \\frac{\text{11 × 10}}{\text{1 × 2}}) × 15 + \\frac{\text{11 × 10 × 9}}{\text{1 × 2 × 3}}) × 6

= {1155 + 825 + 165} × 6

= 2145 × 6 = 12870

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