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Difference between Permutations and Combinations

 

PERMUTATION:

The number of arrangement of r items taken from n distinct items.  Order of arrangement is important in permutation.

nPr = P(n, r) = n!/(n-r)!

 

Question:
How many three letter words can be formed by using the letters of A, B, C, D, E?  Each letter must be used once.

5P3 = 5!/(5-3)! = 60

ABC, ACB, BAC, BCA, CAB, CBA are all different words although the same letters are used.  Therefore all of the words should be considered.

 

ANOTHER WAY TO SOLVE THIS PROBLEM:

First letter can be chosen in 5 ways
Second letter can be chosen in 4 ways
Third letter can be chosen in 3 ways

5 x 4 x 3 = 60 ways

 

  COMBINATION:

The number of selection of r items taken from n distinct items.  Order of selection is not important in combination.

nCr =  n!/[(n-r)! r!]

 

Question:
How many ways can a three member committee be set up by the persons of A, B, C, D, E?

A for Ashley
B for Brandon
C for Carol
D for Daniel
E for Emily

5C3 =  5!/[(5-3)! 3!] = 10

Committee consisting of ABC, ACB, BAC, BCA, CAB, CBA are all the same for the combination.  Therefore, only one will be considered.

 

1

A

B

C

ABC

2

A

C

B

3

B

A

C

4

B

C

A

5

C

A

B

6

C

B

A

7

A

B

D

ABD

8

A

D

B

9

B

A

D

10

B

D

A

11

D

A

B

12

D

B

A

13

A

B

E

ABE

14

A

E

B

15

B

A

E

16

B

E

A

17

E

A

B

18

E

B

A

19

A

C

D

ACD

20

A

D

C

21

C

A

D

22

C

D

A

23

D

A

C

24

D

C

A

25

A

C

E

ACE

26

A

E

C

27

C

A

E

28

C

E

A

29

E

A

C

30

E

C

A

31

A

D

E

ADE

32

A

E

D

33

D

A

E

34

D

E

A

35

E

A

D

36

E

D

A

37

B

C

D

BCD

38

B

D

C

39

C

B

D

40

C

D

B

41

D

B

C

42

D

C

B

43

B

C

E

BCE

44

B

E

C

45

C

B

E

46

C

E

B

47

E

B

C

48

E

C

B

49

B

D

E

BDE

50

B

E

D

51

D

B

E

52

D

E

B

53

E

B

D

54

E

D

B

55

C

D

E

CDE

56

C

E

D

57

D

C

E

58

D

E

C

59

E

C

D

60

E

D

C