MTO MOT There are 2 different arrangements when M is at the beginning.
Giving a total of 6 different arrangements because there are 3 different letters in TOM, all of which can be placed at the beginning.
3 × 2 = 6 different arrangements.
This works for words (where all the letters are different) of any length. If you were doing a 5-letter word for example, the same principle would apply. You could fix the first letter, then rearrange the other 4; giving the number of different arrangements for 4-letter words. But since there are 5 different letters, you multiply by 5. 24 × 5 = 120 different arrangements. So, you can find the number of different arrangements for any word (where all the letters are different) if you know the number of different arrangements for a word (where all the letters are different) with one less letter.
I then realised that 4 × 6 is the same as 4 × 3 × 2 × 1, which can be written using factorial notation, because 4 × 3 × 2 × 1 = 4! (4 factorial).
2! = 2 × 1 = 2 3! = 3 × 2 × 1 = 6
4! = 4 × 3 × 2 × 1 = 24 5! = 5 × 4 × 3 × 2 × 1 = 120
Why factorial is used:
When finding the number of different arrangements for words where all the letters are different you use factorial.
If you start with a 1-letter word
e.g. A
you get 1 arrangement.
As I stated above, to find the number of different arrangements for a 2-letter word you calculate
2 (number of letters in the word) × 1 (number of different arrangements for previous word) = 2!
To find the number of different arrangements for a 3-letter word you do
3 × 2 × 1 = 3!
A 4-letter word
4 × 3 × 2 × 1 = 4!
So, however long the word, you always use factorial because you always multiply the number of letters in the word by the number of different arrangements for a word with one less letter. To find the number of arrangements for a word with one less letter, you take the number of letters in that word and multiply it by the number of different arrangements for a word with one less letter than your second word; and so on, until you get back to 1.
So, by using factorial notation (!) I can predict the number of different arrangements for words where all the letters are different.
6-letter word: 6 × 5 × 4 × 3 × 2 ×1 = 720 different arrangements.
7-letter word: 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 different arrangements.
8-letter word: 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40320 different arrangements.
The formula for finding the number of different arrangements of letters for words/names where all the letters are different is:
n! = a
Where
n = the number of letters in the word/name
a = the number of different arrangements
Now I am going to investigate the number of different arrangements for words/names with one letter repeated twice.
First, I am going to investigate a 2-letter word.
EE
There are 2 letters, both repeated, 1 arrangement.
Now I will investigate a 3-letter word.
BOB BBO OBB
There are 3 letters, 2 repeated, 3 different arrangements.
Now I am going to investigate a 4-letter word.
EMMA MEMA MMAE AEMM
EMAM MEAM MAEM AMEM
EAMM MMEA MAME AMME
There are 4 letters, 2 repeated, 12 different arrangements.
Now I will investigate a 5-letter word.
LAURA ALURA ARLUA ULARA RLAUA
LAUAR ALUAR ARLAU ULAAR RLAAU
LARUA ALRUA ARULA ULRAA RLUAA
LARAU ALRAU ARUAL UALRA RALUA
LAAUR ALAUR ARALU UALAR RALAU
LAARU ALARU ARAUL UARLA RAULA
LUARA AULRA AALUR UARAL RAUAL
LUAAR AULAR AALRU UAALR RAALU
LURAA AURLA AAULR UAARL RAAUL
LRAUA AURAL AAURL URLAA RULAA
LRAAU AUALR AARLU URALA RUALA
LRUAA AUARL AARUL URAAL RUAAL
There are 5 letters, 2 repeated, 60 different arrangements.
Table of Results
From looking at the table of results, I have noticed that the number of different arrangements for a word that has one letter repeated twice is half the number of different arrangements for a word which has the same amount of letters, but none repeated.
For example:
To find the number of different arrangements for EMMA (a 4-letter word with one letter repeated twice), you take the number of different arrangements of a 4-letter word with no letters repeated (e.g. LUCY) and divide it by 2.
24 ÷ 2 = 12 different arrangements.
Why you divide by 2:
When finding the number of different arrangements for words that have one letter repeated twice, you halve the number of different arrangements for a word which has the same amount of letters, but none repeated.
If we look at EMMA (a 4-letter word) for example, it has the same letter (M) repeated twice, and has 12 different arrangements. LUCY (a 4-letter word with no letters repeated) has 24 arrangements.
If we take each M in EMMA to be a different letter, by changing the colour of one of the Ms, we would get 24 arrangements.
EMMA MEMA MEMA AEMM
EMAM MEAM MEAM AEMM
EMMA MMEA MMEA AMEM
EMAM MMAE MMAE AMME
EAMM MAEM MAEM AMEM
EAMM MAME MAME AMME
But for this investigation EMMA and EMMA are the same arrangement because M and M are the same letter (M) and make no difference to the arrangement; they are both the arrangement EMMA.
EMMA and EMMA are both EMMA.
EMAM and EMAM are both EMAM.
EAMM and EAMM are both EAMM.
MEMA and MEMA are both MEMA.
MEAM and MEAM are both MEAM.
MMEA and MMEA are both MMEA.
MMAE and MMAE are both MMAE.
MAEM and MAEM are both MAEM.
MAME and MAME are both MAME.
AEMM and AEMM are both AEMM.
AMEM and AMEM are both AMEM.
AMME and AMME are both AMME.
If each M were different, then for each arrangement there would be another similar arrangement because you could just swap the Ms around (e.g. EMMA → swap Ms around → EMMA), so there are twice the number of arrangements. But since the Ms are the same for the purpose of this investigation, there is only half the number of different arrangements as there would be if the 2 Ms were different letters, so you divide by 2.
I can predict the number of different arrangements for words with one letter repeated twice.
6-letter word: 720 ÷ 2 = 360 different arrangements.
7-letter word: 5040 ÷ 2 = 2520 different arrangements.
8-letter word: 40320 ÷ 2 = 20160 different arrangements.
The formula for finding the number of different arrangements of letters for words/names with one letter repeated twice is:
n! = a
2
Where
n = the total number of letters in the word/name
a = the number of different arrangements
Now I am going to investigate the number of different arrangements for words/names with one letter repeated 3 times.
First, I will investigate a 3-letter word.
CCC
There is only 1 arrangement.
Now I am going to investigate a 4-letter word.
AAAR AARA ARAA RAAA
There are 4 arrangements.
Now I will investigate a 5-letter word.
YYYEL YYYLE YYEYL YYELY YYLEY
YYLYE YEYYL YEYLY YELYY YLYYE
YLYEY YLEYY EYYYL EYYLY EYLYY
ELYYY LYYYE LYYEY LYEYY LEYYY
There are 20 different arrangements.
Table of Results
From looking at the table of results, I have noticed that the number of different arrangements for a word that has one letter repeated 3 times is one sixth of the number of different arrangements for a word which has the same amount of letters, but none repeated.
For example:
To find the number of different arrangements for AAAR (a 4-letter word with 1 letter repeated 3 times), you take the number of different arrangements of a 4-letter word with no letters repeated (e.g. LUCY) and divide it by 6.
24 ÷ 6 = 4 different arrangements.
Why you divide by 6:
When finding the number of different arrangements for words that have one letter repeated 3 times, you take the number of different arrangements for a word which has the same amount of letters, but none repeated, and divide it by 6.
If we look at AAAR (a 4-letter word) for example, it has one letter (A) repeated 3 times, and has 4 different arrangements. LUCY (a 4-letter word with no letters repeated) has 24 arrangements.
If we take each A in AAAR to be a different letter, by changing the colour of 2 of the As, we would get 24 arrangements.
AAAR AAAR AAAR RAAA
AARA AARA AARA RAAA
AAAR AAAR AAAR RAAA
AARA AARA AARA RAAA
ARAA ARAA ARAA RAAA
ARAA ARAA ARAA RAAA
But for this investigation AAAR, AAAR, AAAR, AAAR, AAAR and AAAR are all the same arrangement, because A, A and A are all the same letter (A) and do not change the arrangement; all 6 of the arrangements are AAAR.
AAAR, AAAR, AAAR, AAAR, AAAR and AAAR are all AAAR.
AARA, AARA, AARA, AARA, AARA and AARA are all AARA.
ARAA, ARAA, ARAA, ARAA, ARAA and ARAA are all ARAA.
RAAA, RAAA, RAAA, RAAA, RAAA and RAAA are all RAAA.
If each A were different, then for each arrangement there would be another 5 similar arrangements because you could swap the As around (e.g. AAAR → swap As around → AAAR → swap As around again → AAAR → swap As around again, etc.), so there are 6 times the number of arrangements if each A is different.
But since the As are the same, there are only one sixth the number of different arrangements as there would be if the 3 As were different letters, so you divide by 6.
I can predict the number of different arrangements for words with one letter repeated 3 times.
6-letter word: 720 ÷ 6 = 120 different arrangements.
7-letter word: 5040 ÷ 6 = 840 different arrangements.
8-letter word: 40320 ÷ 6 = 6720 different arrangements.
The formula for finding the number of different arrangements of letters for words/names with one letter repeated 3 times is:
n! = a
6
Where
n = the total number of letters in the word/name
a = the number of different arrangements
Now I am going to investigate the number of different arrangements for words/names with one letter repeated 4 times.
First, I will investigate a 4-letter word.
LLLL
There is only 1 arrangement.
Now I will investigate a 5-letter word.
HHHHU HHHUH HHUHH HUHHH UHHHH
There are 5 arrangements.
Now I am going to investigate a 6-letter word.
SSSSEF SSSSFE SSSESF SSSEFS SSSFSE SSSFES SSESSF SSESFS SSEFSS SSFSSE SSFSES SSFESS
SESSSF SESSFS SESFSS SEFSSS SFSSSE SFSSES
SFSESS SFESSS ESSSSF ESSSFS ESSFSS ESFSSS
EFSSSS FSSSSE FSSSES FSSESS FSESSS FESSSS
There are 30 arrangements.
Table of Results
From looking at the table of results, I have noticed that the number of different arrangements for a word that has one letter repeated 4 times is one twenty-fourth of the number of different arrangements for a word which has the same amount of letters, but none repeated.
For example:
To find the number of different arrangements for SSSSEF (a 6-letter word with 1 letter repeated 4 times), you take the number of different arrangements for a 6-letter word with no letters repeated (e.g. SOPHIE) and divide it by 24.
720 ÷ 24 = 30 different arrangements.
Why you divide by 24:
When finding the number of different arrangements for words that have one letter repeated 4 times, you take the number of different arrangements for a word that has the same number of letters but none repeated, and divide it by 24.
If we look at LLLL (a 4-letter word) for example, the same letter (L) is repeated 4 times, and there is only 1 arrangement. LUCY (a 4-letter word with no letters repeated) has 24 arrangements.
If we take each L in LLLL to be a different letter, by changing the colour of 3 of the Ls, we would get 24 different arrangements.
LLLL LLLL LLLL LLLL
LLLL LLLL LLLL LLLL
LLLL LLLL LLLL LLLL
LLLL LLLL LLLL LLLL
LLLL LLLL LLLL LLLL
LLLL LLLL LLLL LLLL
However, for this investigation all the above arrangements are the same because L, L, L and L are all the same letter (L) and do not change the arrangement; all 24 of the arrangements are LLLL. Since all the Ls are the same there is one twenty-fourth the number of different arrangements as there would be if all the 4 Ls were different letters, so you divide by 24.
I can predict the number of different arrangements for words with one letter repeated 4 times.
7-letter word: 5040 ÷ 24 = 210 different arrangements.
8-letter word: 40320 ÷ 24 = 1680 different arrangements.
The formula for finding the number of different arrangements of letters for words/names with one letter repeated 4 times is:
n! = a
24
Where
n = the total number of letters in the word/name
a = the number of different arrangements
Now that I have found several different formulae for words that have one letter repeated twice, 3 times and 4 times, I am going to try to find a formula that can be used to find the number of different arrangements for words that have one letter repeated any amount of times.
I have already worked out that if you want to find out how many different arrangements there are for a 5-letter word for example, you would calculate 5!. If you wanted to know the number of different arrangements for a 5-letter word with one letter repeated:
-
2× – you would calculate 5! ÷ 2.
-
3× – you would calculate 5! ÷ 6.
-
4× – you would calculate 5! ÷ 24.
This principle applies regardless of the length of the word.
I have noticed that with words that have:
-
1 letter repeated twice, the number by which n! is divided is 2 = 2 × 1 = 2!
-
1 letter repeated 3×, the number by which n! is divided is 6 = 3 × 2 × 1 = 3!
-
1 letter repeated 4×, the number by which n! is divided is 24 = 4 × 3 × 2 × 1 = 4!
As you can see in the previous explanations as to why you divide by 2, 6 and 24, when trying to find the number of different arrangements for a word which has one letter repeated a certain number of times, you take the number of different arrangements for a word which has the same number of letters, but none repeated factorial, and divide it by a certain number. This number that you divide by depends on how many times the one letter is repeated; you divide by the number of times the one letter is repeated factorial.
Why?
If we look at AAAR for example, one letter (A) is repeated 3 times, and there are 4 arrangements. If each A was a different letter, then for each arrangement of letters when the As are the same (i.e. AAAR, AARA, ARAA and RAAA), the As can be rearranged, giving a total of 24 different arrangements. Since there are 3 As, they can be rearranged 6 (3!) different ways. So you take 4 (the number of different arrangements when the As are the same) and multiply it by 3!, to find the number of different arrangements for AAAR if the As were different. So if you want to find the number of different arrangements if the repeated letters are the same, you divide n! by the number of times the one letter is repeated factorial (in the case of AAAR – 3!).
From this, I have worked out a formula to find the number of different arrangements of letters for words/names of any length that have one letter repeated any number of times.
n! = a
r!
Where
n = the total number of letters in the word/name
r = the number of times the same letter is repeated
a = the number of different arrangements
I can now predict the number of different arrangements for any words that have one letter repeated any amount of times.
I can predict that the word XXXXXQZ (a 7-letter word with one letter repeated 5 times) will have 42 different arrangements.
n! = a 7! = 5040 = 42
r! 5! 120
XXXXXQZ XXQZXXX QXZXXXX XXZQXXX
XXXXQXZ XQXXXXZ QZXXXXX XZXXXXQ
XXXXXZQ XQXXXZX XXXXZXQ XZXXXQX
XXXXQZX XQXXZXX XXXXZQX XZXXQXX
XXXQXXZ XQXZXXX XXXZXXQ XZXQXXX
XXXQXZX XQZXXXX XXXZXQX XZQXXXX
XXXQZXX QXXXXXZ XXXZQXX ZXXXXXQ
XXQXXXZ QXXXXZX XXZXXXQ ZXXXXQX
XXQXXZX QXXXZXX XXZXXQX ZXXXQXX
XXQXZXX QXXZXXX XXZXQXX ZXXQXXX
ZXQXXXX ZQXXXXX
There are 42 different arrangements.
I can predict that the word LLLLLV (a 6-letter word with one letter repeated 5 times) will have 6 different arrangements.
n! = a 6! = 720 = 6
r! 5! 120
LLLLLV LLLVLL LVLLLL
LLLLVL LLVLLL VLLLLL
There are 6 different arrangements.
Now that I have investigated the number of different arrangements of letters for words that have one letter repeated any amount of times, I am going to investigate the number of different arrangements of letters and try to find a formula for words that have 2 or more different letters which are repeated (e.g. AABB, CCDD or EEEFFF).
First, I am going to investigate the number of different arrangements of letters for XXYY (a 4-letter word with 2 different letters, each repeated twice).
XXYY YXYX
XYXY YYXX
XYYX YXXY
There are 6 different arrangements.
Now I will investigate a 5-letter word with 2 different letters, 1 repeated 3 times, 1 repeated twice.
XXXYY XYYXX
XXYXY YXXXY
XXYYX YXXYX
XYXXY YXYXX
XYXYX YYXXX
There are 10 different arrangements.
Now I will investigate a 6-letter word with 2 different letters, each repeated 3 times.
XXXYYY XYXYXY YXXXYY YXYYXX
XXYXYY XYXYYX YXXYXY YYXXXY
XXYYXY XYYXXY YXXYYX YYXXYX
XXYYYX XYYXYX YXYXXY YYXYXX
XYXXYY XYYYXX YXYXYX YYYXXX
There are 20 different arrangements.
Now I will investigate another 6-letter word with 2 different letters, this time with 1 repeated 4 times, 1 repeated twice.
XXXXYY XXYYXX YXXXXY
XXXYXY XYXXXY YXXXYX
XXXYYX XYXXYX YXXYXX
XXYXXY XYXYXX YXYXXX
XXYXYX XYYXXX YYXXXX
There are 15 different arrangements.
From looking at my results, I have noticed a pattern and found a formula. The number of different arrangements of letters for any word/name can be found by taking n!, and dividing it by the product of the number of repeats for each different letter factorial.
For example:
To find the number of different arrangements for XXXXYY (a 6-letter word with 2 different letters, 1 repeated 4 times, 1 repeated twice), you do:
total number of letters factorial = 6! = 15
number of Xs factorial × number of Ys factorial 4! × 2!
If you have 2 or more different letters which are repeated the same principle as in the explanation to the formula n! / r! = a applies. However, because there is more than just one letter repeated, instead of just dividing by the number of times the one letter is repeated factorial, you divide by the product of each different letter’s number of repeats factorial. You multiply them together because if there is more than one letter repeated, the other letter(s) repeated can also be rearranged, giving more arrangements, depending on how many times the other letter(s) is repeated.
If we look at XXYY for example, there are 6 different arrangements because both Xs are the same and both Ys are the same.
XXYY XYXY XYYX YXYX YYXX YXXY
But if the Xs and Ys were different there would be more arrangements. Firstly, you could rearrange the Xs so there would then be twice as many arrangements.
XXYY XYXY XYYX YXYX YYXX YXXY
XXYY XYXY XYYX YXYX YYXX YXXY
But because another letter is repeated (Y), for each of the 12 arrangements the Ys can also be rearranged. So you have to multiply the number of different ways the 2 Ys (if they were different) can be arranged (found by using n! = a) by 12 to get 24 different arrangements.
XXYY XYXY XYYX YXYX YYXX YXXY
XXYY XYXY XYYX YXYX YYXX YXXY
XXYY XYXY XYYX YXYX YYXX YXXY
XXYY XYXY XYYX YXYX YYXX YXXY
This is why you have to multiply the number of repeats factorial for each different letter together.
The formula for finding the number of different arrangements of letters for any words/names is:
n! = a
x! y!
Where
n = the total number of letters in the word/name
x = the number of times one letter is repeated in the word/name
y = the number of times another letter is repeated in the word/name
a = the number of different arrangements
Using this formula, I can predict the number of different arrangements of letters for any word/name.
I predict that the number of different arrangements for the word XXYYZ is:
n! = a 5! = 30
x! y! z! 2! × 2! × 1!
XXYYZ XXYZY XXZYY XYXYZ XYXZY XYYXZ
XYYZX XYZXY XYZYX XZXYY XZYXY XZYYX
YXXYZ YXXZY YXYXZ YXYZX YXZXY YXZYX
YYXXZ YYXZX YYZXX YZXXY YZXYX YZYXX
ZXXYY ZXYXY ZXYYX ZYXXY ZYXYX ZYYXX
There are 30 different arrangements.
I predict that the number of different arrangements for the word WWXYZ is:
n! = a 5! = 60
w! x! y! z! 2! × 1! × 1! × 1!
WWXYZ WXZWY WZXWY XYWWZ YWZWX ZWXWY
WWXZY WXZYW WZXYW XYWZW YWZXW ZWXYW
WWYXZ WYWXZ WZYWX XYZWW YXWWZ ZWYWX
WWYZX WYWZX WZYXW XZWWY YXWZW ZWYXW
WWZXY WYXWA XWWYZ XZWYW YXZWW ZXWWY
WWZYX WYXZW XWWZY XZYWW YZWWX ZXWYW
WXWYZ WYZWX XWYWZ YWWXZ YZWXW ZXYWW
WXWZY WYZXW XWYZW YWWZX YZXWW ZYWWX
WXYWZ WZWXY XWZWY YWXWZ ZWWXY ZYWXW
WXYZW WZWYX XWZYW YWXZW ZWWYX ZYXWW
There are 60 different arrangements.
I have found a formula that can be used to find the number of different arrangements of letters for any word or name, explained and proved why it works, and have achieved the aim of my investigation.
By Thomas Tam 10B1
