Combinations in Statistics: Formula, Examples, and Applications, Essays (high school) of Chemistry

Download Combinations in Statistics: Formula, Examples, and Applications and more Essays (high school) Chemistry in PDF only on Docsity! ALDERSGATE COLLEGE STATISTICS 9 HIGH SCHOOL LESSON 4.3Combinations Combination is an arrangement of objects/things without reference to theorder in which they are arranged. Thus, abc ,bac , cba are all combinations, althoughthey are different permutation.The number of possible combinations of r objects from a collection of nobjects is given by the formula C r= n! (n−r ) !r !n ❑ Note that: C r= ¿n ❑ Cn−r , Cn=1 , C0= ¿n ❑ 1n ❑ n ❑ ¿¿ Example 1 Evaluate the following.a) C8❑ 3 b) C12❑ 0 c) C15❑ 8 Solution a) C8❑ 3= 8 ! (8−3 )!3 ! = 8! 5 !3 ! = 8 ∙7 ∙6 ∙5 ! 5 !(3 ∙2∙1) =8 ∙7 ∙6 6 =56 b) C12❑ 0= 12! (12−0 )! 0! = 12! 12 ! (1) =1 c) C15❑ 8= 15! (15−8 ) !8 ! = 15 ! 7 !8 ! =15 ∙14 ∙13 ∙12 ∙11 ∙10 ∙9 ∙8! (7 ∙6 ∙5 ∙4 ∙3 ∙2 ∙1 )8 ! =13 ∙11∙9 ∙5 1 =6435 Example 2 The poker hand consists of five cards dealt from an ordinary deck of 52playing cards.a) How many possible poker hand are there?b) How many different hands are there consisting of three aces andtwo kings?c) How many different hands are there consisting of all red cards?d) How many different hands are there consisting of 2 hearts, 2diamonds and 1 spade? Solution There are 52 playing cards, therefore n=52. a) C52❑ 5= 52 ! (52−5 )!5 ! = 52! 47 ! ∙3 ! =2598960 b) for three aces: C4 ❑ 3 for two kings: C4 ❑ 2 Using the fundamental principle of counting we have C4❑ 3 ∙ C4 ❑ 2=24ways.c) Since there are 26 red cards, n=26. C26 ❑ 5= 26 ! (26−5 ) !5 ! = 26 ! 21!5 ! =26 ∙25 ∙24 ∙23 ∙22 ∙21! 21 !(5∙4 ∙3 ∙2∙1) =65780 d) for two hearts: C13 ❑ 2 for two diamonds: C13 ❑ 2 1 | M O D U L E 4 : C O U N T I N G T E C H N I Q U E S ALDERSGATE COLLEGE STATISTICS 9 HIGH SCHOOL for one spade: C13 ❑ 1 Using the fundamental principle of counting we have C13 ❑ 2 ∙ C13 ❑ 2 ∙ C13 ❑ 1=79092 2 | M O D U L E 4 : C O U N T I N G T E C H N I Q U E S