 # Book 3: Permutation Combination

Sampling, replacement, reordering, cards, die, coins, sequences

Permutation combination is considered a hard 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 Math 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.

### Concepts Covered

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.

Here is a typical solved example 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?

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 $$times$ 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 $\times$ 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*$\frac {15*14} {1*2}$+$\frac {11*10} {1*2}$ * 15 + $\frac {11*10*9} {1*2*3}$} x 6

= {1155 + 825 + 165} * 6

= 2145 * 6 = 12870

### Chapterwise details of Wizako's Math Lesson Books

Math Lesson Book 1

 1 Linear Equations Details 2 Quadratic Equations Details 3 Set Theory Details 4 Sequences & Series Details 5 Number Properties & Theory Details 6 Inequalities Details 7 Functions

Math Lesson Book 2

 1 Descriptive Statistics Details 2 Ratio Proportion Details 3 Mixtures 4 Interest 5 Rates: Speed Distance Details 6 Rates: Races Details 7 Rates: Work Time Details

Math Lesson Book 3

 1 Percents Details 2 Profits Details 3 Permutation Combination Details 4 Probability Details 5 Geometry Details 6 Solid Geometry 7 Coordinate Geometry Details