Dupligrams

13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 *3 *2 * 1

6,227,020,800 / (2 * 1)(4 * 3 * 2 * 1)(2 * 1)  = 6,227,020,800 * 96 = 64,864,800

16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1 = 20,922,789,888,000 / (4*3*2*1)(2*1)(2*1)(3*2*1) = 36,324,288,000

Number of arrangements= 11*10*9*8*7*6*5*4*3*2=

Two duplicates: E,T

Distinguishable Arrangements= Answer to How many arrangements / (2*1)(2*1)=

My celebrity is Jessica Simpson.

Number of arrangements: 14*13*12*11*10*9*8*7*6*5*4*3*2*1 = 87,178,291,200

Duplicates: There are duplicates of S's and I's

Repititions: There are 4 S's and 2 I's [(4*3*2*1) * (2*1)]

# of distinguishable arrangements: 87,178,291,200 / 48 = 1,816,214,400

Since you need to remove the possibilities of distinguishable arrangements of the repetitions, you need to divide their possibilities from the whole. In other words, if you have the word Loop, you would calculate the original arrangement as 4 x 3 x 2 x 1, equaling 24. But, you need to take out the possibilities of the O's in the combination. Their arrangement is calculated as 2 x 1 = 2, since there are two of them. Then, you divide 24 (Loop's original arrangement, 4 x 3 x 2 x 1) by 2 (O's combinations, 2 x 1), equaling 12.

The same formula is applied to TTTFTTFTT. The original combination would be 9x8x7x6x5x4x3x2x1 = 362,880. T's repetitions factor as 5040, and F's factor as 2. Multiply those to get 10,080. Then you divide the original arrangement by 10,080. 362,880/10,080, which equals 36.

Bubba - Phoenix claimed #5 first. Do SAN FRANCISCAN
Also tell us how you derived your celebrity count.

GolfGirl - Tell us how you got 6360

BassLady - be sure to spell MASSACHUSETTS correctly

Houdini - I erased yours so BassLady could work on it. 7 points!

If I did not mention your solution, I did not find anything wrong with it
yet...

15.

n= n*(n-1)*n-2*(n-3)....*(n-n+1)

ARRANGEMENTS= 8*7*6*5*4*3*2*1=40320

DUPLICATES: H: 6*5*4*3*2*1=720

                  T: 2*1=2

DISTINGUISHABLE ARRANGEMENTS: 40320/720/2=56

To find the number of distinguishable arrangements, you must divide the number of total arrangements by ALL of the duplicates as shown above.

ARRANGEMENTS= 13*12*11*10*9*8*7*6*5*4*3*2*1=6227020800

DUPLICATES: H: 3*2*1=6

                  T: 2*1=2

DISTINGUISHABLE ARRANGEMENTS: 6227020800/6/2=518918400

#14 ALABAMA

7*6*5*4*3*2*1=5040 WAYS

(7*6*5*4*3*2*1)/(4*3*2*1)=5040/24=210

JOHNNY COCHRAN

13*12*11*10*9*8*7*6*5*4*3*2*1=6227020800

(13*12*11*10*9*8*7*6*5*4*3*2*1)/(2*1)(2*1)(3*2*1)(2*1)=6227020800/48=129729600

#20

I will finish the rest.

#9 LOOLOO

There are 6 letters in the word LOOLOO.  To find the total number of arrangements regardless of repetitions, we would take 6*5*4*3*2*1 (with their being 6 letters that could fit in the first possible slot, 5 for the second, 4 remaining for the 3rd and so on).  We would then need to divide by the total number of possible repetitions in order to find how many different distinguishable arrangements there are.  For the letter O that would be 4*3*2*1 and for the L that would be 2*1.  This would look like (6*5*4*3*2*1)/((4*3*2*1)(2*1)) or 5040/48 or 105 distinguishable arrangements.

 So basically, the rule is n(n-1)(n-2)(n-3)../r(r-1)(r-2)(r-3).. where n = number of letters and r = number of repetitions.

 Using the same logic, we can check James Cagney

We determine the number of arrangements which is 11*10*9*8*7*6*5*4*3*2*1.  There are two repeating letters (A & E) which gives us 2*1 and 2*1 using the formula above.  This leaves us with 11*10*9*8*7*6*5*4*3*2*1/((2*1)(2*1)) or 39,916,800/4 or 9,979,200.

     11*10 *9*8*7*6*5*4*3*2*1 =39,916,800

I got the distinguishable arrangement by using n(n-1)(n-2) which is 39,916,800(39,916,800-1)(39,916,800-2)

.

#14 Alabama 

7*6*5*4*3*2*1 / 4*3*2*1=210  Rule: If n= #of letters and k=repetition of a letter then
 n(n-1)(n-2)....(n-k+1) is your number of distiguished arangements. 

8*7*6*5*4*3*2*1 / 2*1= 20160

I choose#13.ABAABAA

The equation that I used to figure this out was: (5x4x3x2x1)(2x1)=240

(7x6x5x4x3x2x1)= 5040 which equals the number of arrangements then I divided that number by the number above to find the distinguishable arrangements :)

So, 5040/240=21

I figured this problem out similiarly to the one above also.

I did (9x8x7x6x5x4x3x2x1)=362880 which is the number of arrangments

Then to find the distinguishable arrangments I divided 362880 by (2x1)=181440 which is the distinguishable arrangments.

WORD 
NUMBER 
HOW MANY 
HOW MANY 
NUMBER OF
DISTINGUISHABLE 
NAME
of LETTERS 
ARRANGEMENTS 
DUPLICATES 
REPETITIONS 
 ARRANGEMENTS
 EMEE
 4
 24
 1
 E-3
 4
 William Carlos Williams
 21
 5.109094217^19
6
 W-2 I-4 L-5 A-3 M-2 S-2
 3.695814683^14

#5   4 letters 4*3*2*1=24   arrangements        because there are 3 E's. 3*2*1=6

    Distinguishable arrangements  24/6=4  

I choose #5. EMEE

WORD 
NUMBER 
HOW MANY 
HOW MANY 
NUMBER OF
DISTINGUISHABLE 
NAME
of LETTERS 
ARRANGEMENTS 
DUPLICATES 
REPETITIONS 
 ARRANGEMENTS
 EMEE
 4
 24
 3:E
 E 3
 4

(4*3*2*1)/3*2*1)=4

D=N/R   The formula for number of distinguishable arrangements is using the factorial product of the number divided by the product of all factorial products of the repetitions.

My celebrity is Mariah Carey

WORD 
NUMBER 
HOW MANY 
HOW MANY 
NUMBER OF
DISTINGUISHABLE 
NAME
of LETTERS 
ARRANGEMENTS 
DUPLICATES 
REPETITIONS 
 ARRANGEMENTS
 Mariah Carey
 11
 39,916,800
 2: A R
 3A  2R
 3,326,400

(11*10*9*8*7*6*5*4*3*2*1)/(3*2*1)(2*1)=3,326,400

You will notice that in this case "Bill" can be changed 24 ways.  But notice that there are some duplicates.   actually with this four letter word whith one letter repeating once, there is only 12 ways to arrange "Bill" and still get something different. 

the equasion in this case would be (4*3*2*1)/(2*1)=12distinguishable arrangements

Now, onto Chaka

English-
As we learned in Anograms, the word like Chaka Kahn would be (9*8*7*6*5*4*3*2*1)
but we must account for the repetitions which, in this case are represented by (3*2*1)(2*1)(2*1)

so-   (9*8*7*6*5*4*3*2*1)/(3*2*1)(2*1)(2*1)=15120 distinguishable arrangements

#7 Bibbi

Arrangements:
5*4*3*2*1=120   r=(2*1)(i) r=(3*2*1)(b)
5*4*3*2*1/(2*1)(3*2*1)= 120/12 = 10 total arrangements

Arrangements: 10*9*8*7*6*5*4*3*2*1=3628800     r=(2*1) (o)
10*9*8*7*6*5*4*3*2*1\2=1814400

 #2 Bob

Duplicates= 1 B

Repetitions = 2 B's

Arrangements = 6/1/2= 3 distinguishable arrangements

Part one assignment

R= (11x10x9x8x7x6x5x4x3x2x1)= 39916800
R= (4x3x2x1)(4x3x2x1)(2x1)= (24)(24)(2)= 1152
=39916800/1152=  34650 arrangements

I choose #11: LOLOLO

For LOLOLO, there are 6 letters meaning the arrangements would equal 6*5*4*3*2*1, which is 720.

For any word, the possible arrangements can be found by:

1. Identifying n (n=number of letters in word)

2. Calculating n*(n-1)*(n-2)*(n-3)... until n-?=1

3. The resulting product is the number of possible arrangements.

When letters are repeated, you perform steps 1-3 above and then continue to the steps below:

4. Identify p, q, r, s, t... (each variable aligns with a repeated letter)

5. Calculate p*(p-1)*(p-2)*(p-3)... until p-?=1 for each of the variables identified in step 4

6. Multiply all of the products found in steps 4 and 5 together

7. Divide the product in step 3 by the product found in step 6 for the possible number of distinguishable arrangements.

For Sandra Bullock:

There are 13 letters in her name.  A and L are both repeated twice.

Therefore:

(13*12*11*10*9*8*7*6*5*4*3*2*1) = 6,227,020,800 arrangements

(2*1) * (2*1) = 4  (this is 2*1 for both A and L)

6,227,020,800 / 4 = 1,556,755,200 distinguishable arrangments

#10

WORD 
NUMBER 
HOW MANY 
HOW MANY 
NUMBER OF
DISTINGUISHABLE 
NAME
of LETTERS 
ARRANGEMENTS 
DUPLICATES 
REPETITIONS 
 ARRANGEMENTS
OOOLOO 
 6
 720 
 1: o
 5
 6

This is actually one of the simpler words, with only one letter repeated, and one unique letter outsied that.  The solution is:

(6*5*4*3*2*1)/(5*4*3*2*1) = 720/120 = 6

Celebrity Name is Daisy Fuentes.

WORD 
NUMBER 
HOW MANY 
HOW MANY 
NUMBER OF
DISTINGUISHABLE 
NAME
of LETTERS 
ARRANGEMENTS 
DUPLICATES 
REPETITIONS 
 ARRANGEMENTS
Daisy Fuentes 
 12
 479001600
 2: e,s
2 E, 2 S 
 119750400

(12*11*10*9*8*7*6*5*4*3*2*1)/(2*1)(2*1) = 479001600/4 = 119750400

Good looking out superduke! thanks!

The formula for number of Distinguishable Arrangements is N/R, where:
N is the total number of arrangements if repetition of certain letters is allowed,
N = n(n − 1)(n − 2)……….(1)  (
R is the total number of arrangements of repeated letters
R = (n1*n1*........)(n2*n2*...........)......(nk*nk*....)

In my example for LOOOLLO:  You calculate this by using the factoral of 7 for the number of letters in the word and dividing that by the factoral of repeated letters
7*6*5*4*3*2*1= 5,040
“L” is repeated 3 times so: 3*2*1= 6
“O” is repeated 4 times so: 4*3*2*1= 24
6*24= 144
5,040/144= 35

In my celebrity name, Russell Crowe, example:
12*11*10*9*8*7*6*5*4*3*2*1 =479,001,600
“R” is repeated 2 times so:  2*1=2
“S” is repeated 2 times so: 2*1=2
“E” is repeated 2 times so: 2*1=2
“L” is repeated 2 times so: 2*1=2
2*2*2*2= 32
479,001,600/32= 1,442,775.9