Investigate the number of different arrangements of the letters in a name.

8. RMKA

9. RMAK

10. KRMA

11. AKRM

12. MAKR

13. AMKR

14. RAMK         24 WAYS

15. KRAM

16. MKRA

17. KMRA

18. AKMR

19. RAKM

20. MRAK

21. MRKA

22. AMRK

23. KAMR

24. RKAM

1. JAMES
2. SJAME
3. ESJAM
4. MESJA
5. AMESJ
6. AMEJS
7. SAMEJ
8. JSAME
9. EJSAM
10. MEJSA
11. AMEJS
12. AMJES
13. SAMJE
14. ESAMJ
15. JESAM
16. MJESA
17. MJEAS
18. SMJEA
19. ASMJE
20. EASMJ
21. JEASM
22. JEAMS
23. SJEAM

                                                                                                                    Page 4

24. MSJEA
25. AMSJE
26. EAMSJ
27. EAMJS
28. SEAMJ
29. JSEAM
30. MJSEA
31. AMJSE
32. EAMJS
33. SEAMJ
34. JSEAM
35. MJSEA
36. MJSAE
37. EMJSA
38. AEMJS
39. SAEMJ
40. JSAEM
41. MJSAE
42. EMJSA
43. AEMJS
44. SAEMJ
45. SAEJM
46. MSAEJ
47. JMSAE
48. EJMSA
49. AEJMS
50. AEJSM
51. MAEJS
52. SMAEJ
53. JSMAE
54. EJSMA
55. JESMA
56. AJESM
57. MAJES
58. SMAJE
59. ESMAJ
60. ESMJA
61. AESMJ
62. JAESM
63. MJAES
64. SMJAE
65. MSJAE

                                                                                                                          Page 5

66. EMSJA
67. AEMSJ
68. JAEMS
69. SJAEM
70. SAJEM
71. MSAJE
72. EMSAJ
73. JEMSA
74. AJEMS
75. AJESM
76. MAJES
77. SMAJE
78. ESMAJ
79. JESMA
80. JSEMA
81. AJSEM
82. MAJSE
83. EMAJS
84. SEMAJ
85. SEMJA
86. ASEMJ
87. JASEM
88. MJASE
89. EMJAS
90. MEJAS
91. SMEJA
92. ASMEJ
93. JASME
94. EJASM
95. JMASE
96. EJMAS
97. SEJMA
98. ASEJM
99. MASEJ
100. MASJE
101. EMASJ
102. JEMAS
103. SJEMA
104. ASJEM
105. ASJME
106. EASJM
107. MEASJ

                                                                                                          Page 6

108. JMEAS
109. SJMEA
110. SJMAE
111. ESJMA
112. AESJM
113. MAESJ
114. JMAES
115. JAMSE
116. EJAMS
117. SEJAM
118. MSEJA
119. AMSEJ

      120.          MAJES

These names all have two letters the same:

                                                                    1.JJ

                                                                    2.BOB

                                                                    3.SAMM

                                                                    4. SARAH

                                                                     

1. JJ         1 WAY

1. BOB
2. BBO    3 WAYS
3. OBB

1. SAMM
2. MSAM
3. MMSA
4. AMMS
5. AMSM
6. MAMS       12 WAYS
7. SMAM
8. MSMA
9. SMMA
10. ASMM
11. MASM
12. MMAS

                                                                                                                      Page 7

1. SARAH
2. HSARA
3. AHSAR
4. RAHSA
5. ARAHS
6. ARASH
7. HARAS
8. SHARA
9. ASHAR
10. RASHA
11. RASAH
12. HRASA
13. AHRAS
14. SAHRA
15. ASAHR
16. ASARH
17. HASAR
18. RHASA
19. ARHAS
20. SARHA
21. ASRHA
22. AASRH
23. HAASR
24. RHAAS
25. SRHAA
26. RSHAA
27. ARSHA
28. AARSH
29. HAARS
30. SHAAR
31. SHARA
32. ASHAR
33. RASHA
34. ARASH
35. HARAS
36. HARSA
37. AHARS
38. SAHAR
39. RSAHA
40. ARSAH
41. ASRHA
42. AASRH

                                                                                                                  Page 8

43. HAASR
44. RHAAS
45. SRHAA
46. SHRAA   6O WAYS
47. ASHRA
48. AASHR
49. RAASH
50. HRAAS
51. HRAAS
52. SHRAA
53. ASHRA
54. AASHR
55. RAASH
56. RAAHS
57. SRAAH
58. HSRAA
59. AHSRA
60. AAHSR

These names all have three letters the same:

                                                                      1.

                                                                      2. MMM

                                                                           3. EMMM

                                                                           4. EAMMM

      1. MMM   1 WAY

1. EMMM
2. MEMM
3. MMEM   4 WAYS
4. MMME

1. EAMMM
2. MEAMM
3. MMEAM
4. MMMEA
5. AMMME
6. AMMEM
7. MAMME
8. EMAMM

                                                                                                            Page 9

9. MEMAM
10. MMEMA
11. MEMMA
12. AMEMM
13. MAMEM   20 WAYS
14. MMAME
15. EMMAM
16. EMMAM
17. EMMMA
18. AEMMM
19. MAEMM
20. MMAEM

Here is a table to show my results:

To discover the number of different ways for a name with all different letters in it you use the formula n! For example 4! = 24. To show how this works you can use the formula n! = n (n-1) (n-2) (n-3) (n-4) (n-5) etc. This means that you times n by all the numbers beneath it. So for 4! You do 4 x (4-1) (4-2) (4-3) which is                      4 x 3 x 2 x 1. So 4 x 3 x 2 x 1 = 24.

Example:

               N = 5.  So you would times 5 by all the numbers beneath it. So it would be 5 x (5-1) (5-2) (5-3) (5-4) which is 5 x 4 x 3 x 2 x 1 which = 120 and is what I found in my results.

Using this method I can predict the number of different ways for all letters different with 6 letters in and 7 letters in without writing it up.

So if n = 6 it would be 6 (6-1) (6-2) (6-3) (6-4) (6-5) which is 6 x 5 x 4 x 3 x 2 x 1.

So 6! Would be 720. If n = 7 it would be 7 (7-1) (7-2) (7-3) (7-4) (7-5) (7-6) which is 7 x 6 x 5 x 4 x 3 x 2 x 1. So 7! Would be 5040.

Another formula to find the number of different arrangements with a word with all different letters in is N! = N (N-1)! Which means to find the number of arrangements you change N! with the number of letters in the word. For example

 

                                                                                                                   Page 10

                                                                                                                     

An 8 letter word with all letters different would be 8 = 8 (result for 8-1)! So it would be 8 = 8 (8-1)! So it would be 8 = 8 (5040) which would be 8 = 8x5040 which is 40320.                                      

To find the number of different ways for a name that hasn’t got all letters different you use the formula N!

                                 D!

N = The number of letters and D = The number of letters the same

So if N = 4 and D = 2 you would place them into the formula. So it would be 4!

                                                                                                                            2!

So it would be 4 (4-1) (4-2) (4-3)  which is 4 x 3 x 2 x 1

                            2 (2-1)                                     2 x 1

so it would be 24  which is 12 and is what my results show.

                        2

Another example is if N = 5 and D = 3

When the numbers are placed into the formula it becomes 5!

                                                                                             3!

Which is 5 (5-1) (5-2) (5-3) (5-4)  which is 5 x 4 x 3 x 2 x 1

                 3 (3-1) (3-2)                                        3 x 2 x 1

so it would be 120  which is 20 and is also what my results show.

                         6

Using this method I am able to predict the amount of different ways there are for names that haven’t got all letters the same. For example if I wanted to find the number of ways for a six letter word with two letters the same I would do 6!

                                                                                                                      2!

So it would be 6 (6-1) (6-2) (6-3) (6-4) (6-5)  which is 6 x 5 x 4 x 3 x 2 x 1

                                        2 (2-1)                                               2 x 1

So it would be 720. The number of ways for a six letter word with two letters the same would be 360. Another prediction would be if N = 7 and D = 3 I would place them in the formula so it would be 7!  Which is 7 (7-1) (7-2) (7-3) (7-4) (7-5) (7-6)

                                                        3!                               (3-1) (3-2)

So it would be 5040  so the number of ways for a seven letter word with three

                          6

letters the same would be 840.

In conclusion using the formulas N! and N! make it easier to find the number of different ways of arrangements in a word.