Counting: Sequences, Permutations, and Combinations, Summaries of Mathematics

Download Counting: Sequences, Permutations, and Combinations and more Summaries Mathematics in PDF only on Docsity!

COUNTING

Mathematics for Computing

(IT 1030)

SEQUENCES

•What is a Sequence? •A Sequence is a list of things (usually numbers) that are in order; Infinite or Finite

SET VS. SEQUENCE

Ex: {0, 1, 0, 1, 0, 1, ...} is the sequence of alternating 0s and 1s. The set is just {0,1} or {1,0}. Set Sequence Terms need not to be in order Terms must be in order Values cannot repeat Values can repeat

ARITHMETIC SEQUENCE

 It has a common difference between successive terms. Ex: 2, 4, 6, 8, … Q: Find the 10 th term and the sum of first 10 terms of the following sequence A n . A n : {3, 8, 13, 18, 23,…}

SIGMA NOTATION

 It represents summation of many similar terms. Q: Expand the followings (i) (ii)

PROPERTIES OF SIGMA NOTATION

 There are a couple of formulas for summation notation. Show that,

FACTORIAL(N)

 Find the following values (i) 3! (ii) 5! * 2! (iii) 0!

AND NOTATIONS

Find the followings,

EXAMPLE

(i) How many ways can the letters in the word COMPUTER be arranged in a row? All the eight letters are in the word COMPUTER are distinct, so the number of ways, 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40, (ii) How many ways can the letters in the word COMPUTER be arranged if the letters “CO” must remain next to each other (in order) as a unit? 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040.

PERMUTATIONS OF SELECTED ELEMENTS

 If n and r are integers and 1<=r<=n, then the number of r permutations of a set of n elements is given by the formula  Example: A license plate begins with three letters. If the possible letters are A, B, C, D and E, how many different permutations of these letters can be made if no letter is used more than once? (Ans: 60 )

QUESTION

(i) 16 teams enter a competition. They are divided up into four Pools (A, B, C and D) of four teams each. Every team plays one match against the other teams in its Pool. After the Pool matches are completed:

• the winner of Pool A plays the second placed team of Pool B
• the winner of Pool B plays the second placed team of Pool A
• the winner of Pool C plays the second placed team of Pool D
• the winner of Pool D plays the second placed team of Pool C The winners of these four matches then play semi-finals, and the winners of the semi-finals play in the final. How many matches are played altogether?

QUESTION

 How many “Mahajana sampatha” Tickets can be printed in a single draw ?? (numbers are selected from 0 to 9 and it can repeat)

The End