2 Pairs Of Same Number example essay topic

Firstly we arrange EMM As Name. 1) E AMM 7) MEM 2) EXAM 8) MAME 3) MEM A 9) A MME 4) MEAN 10) A EMM 5) MME A 11) A MEM 6) MMA E 12) EMMA Secondly we arrange lucy's name. 1) Lucy 12) C yul 22) Yul c 2) Luc 13) Culy 23) Y cul 4) Lycus 14) Culy 24) Yluc 5) Lc uy 15) Cyl u 25) U cyl 6) Lc yu 16) Clu 7) Ulc y 17) Curl 8) Ugly 18) Yluc 9) Ucl 19) Y ucl 10) Uly c 20) Y clu 11) Ulc 21) Yl cu From these 2 investigation I worked out a method: Step 1: 1234-Do the last two number first then you get 1243.1243-Do the last three numbers and try the possibility. 1423.1432. 1342.1324, because the number 2 has been the first number of last three numbers, so we dont do it again. Step 2: we have list all arrangements of 1 go front, so we do 2 go front.

We are trying to work out a formula which can calculate the number of arrangement when we look at a number. Carry on, if a number has 6 figure, then the total of arrangement should be 120 times by 6, and get 720, and 720 is the total of arrangements. Carry on, if a number has 7 figure, then the total of arrangement should be 720 times by 7, and get 5040, the total of arrangement is 5040. This is my prediction, lets work it out a formula, and confirm it. 3 figure with different number it has 6 arrangements 4 4 6 5 4 6 5 6 4 6 5 6 7 4 6 5 6 7 8 4 6 5 6 7 8 so on We can rewrite it as: 1 fig 1 2 fig 1 2 3 fig 1 2 3 4 fig 1 2 3 4 5 fig 1 2 3 4 5 6 fig 1 2 3 4 5 6 so on Theres a symbol for the frequency above, thats I. For example: 1 2 = 2 i 1 2 3 = 3 i 1 2 3 4 = 4 i so on So if n represent the number of figures of a number, then it has arrangements of ni.

The formula: NI NI: Can be calculated on calculator. Process: pres key N (the number of figure), then press key I, then you would get the arrangements. 4 fig, one arrangement. a = n/24 = (1 2 3 4) /24 = 1 the formular works try 5 figures 11112 11121 11211 -- 5 arrangements 12111 21111 a = n/24 = (1 2 3 4 5) /24 = 5 the formular works So formula is confirmed Lets investigate the formula, and improve it 24 110 so the formula for this is x = ni so if A represent arrangement, and n represent numbers of figures, x represent the number fo same number, and the formula is: a = ni / xi notice I can not be cancel out. What about if a number has 2 pairs of same number. what would happen to the formula.

Lets try 4 fig with 2 pairs of 2 same number. For example: 11122, 111122 lets try if the formula still work. The formula is a = ni / xixi but we need to change the formula, because there are 2 pairs of same numbers with different number of figures. so we change the formula to a = ni / x 1 i x 2 i Lets try 5 figures with 3 same number, and 2 same number. Lets try 7 fig, with 3 same number, and 4 same number. The formular is confirmed.

What about three pairs of same number The formular need to be rewritten as a = ni / xixixi There are three xi need to mutiply ni, because there are three pairs of same number. if there are two pair of same number of figures of same number, then there are only two xi need to mutiply, and if there are two pair of different number of figures of same number, then there would be x 1 i and x 2 i need to mutiply. Formular is confirmed What about three pairs of different number of figures of a number For example: 122333 according the formula, the total arrangment is a = (1 2 3 4 5 6) / (1 1 2 1 2 3) = 60 Lets confirm it: 122333 212333 231332 3 -- - 123233 213233 232133 so on -- - 30 arrangements 123323 213323 232313 123332 213332 232331 -- 30 arrangements 132233 221333 233123 132323 223133 233132 132332 223313 133213 133223 223331 233231 133232 231233 233312 133322 231323 233321 The formular works Formular is confirmed From the investigation above we find out the formular for calculating the number of arrangements, its a = ni / xi a represent the total arrangements n represent the number of figures of the number I represent the key I x represent the numbers of figures of same number of the number if there are more than one pair of same number, x 2, or x 3, so on may added to the formular, it depend how many pairs of same number. For example: for 2 pairs of same number of figures of same number of a number the formula is a = ni / xixi for 2 pairs of different number of figures of same number of a number the formula is a = ni / x 1 ix 2 i for 3 pairs of same number of figures of same number of a number the formula is a = ni / xixixi form 3 pairs of different number of figures of same number of a number the formular is a = ni / x 1 ix 2 ix 3 i. The formular can be also used to the arrangements of letter. Use this formular, we can find out the total arrangements of all numbers and letters.