Question

How many different arrangements of the letters in the word BETTER are there?

Answer 1

7!/3!

Answer 2

4^6=4096

Answer 3

hmmm

Answer 4

haven't seen the right answer yet

Answer 5

isn't it 4096?

Answer 6

the choices are 180,360, and 720

Answer 7

if the letters were all different, it would be \(6!\)

Answer 8

no

Answer 9

yeahh

Answer 10

but since you cannot tell apart the two "E" and two "T" it is
\[\frac{6!}{2\times 2}\]

Answer 11

but the letters have multiples

Answer 12

360

Answer 13

4096?

Answer 14

\[\frac{6\times 5\times 4\times 3\times 2}{4}=6\times 5\times 3\times 2\]

Answer 15

im confused

Answer 16

i bet

Answer 17

me too

Answer 18

lets go slow

Answer 19

okay

Answer 20

if all the letters were different (they are not, but i am pretending) there would be
6 choices for the first letter
5 for the second
4 for the third
3 for the fourth
2 for the fifth and
1 for the sixth

Answer 21

by the counting principle the number of ways to do this is found by multiplying all the choices together, i.e.
\[6\times 5\times 4\times 3\times 2\times 1=6!\]

Answer 22

so it is 360

Answer 23

but in this case you have repeated letters
there are two E and two T so you have overcounted

Answer 24

you have to divide by 2 twice, because of this

Answer 25

why do u divide by 2 twice?

Answer 26

so the answer is
\[\frac{6\times 5\times 4\times 3\times 2}{4}=6\times 5\times 3\times 2\]

Answer 27

because there are two E and also two T

Answer 28

i get 180

Answer 29

i get all of it except why divide by 4

Answer 30

could u go into depth

Answer 31

because for example
\[E_1BE_2T_1T_2R\] looks the same as
\[E_2BE_1T_1T_2R\] looks the same as
\[E_1BE_2T_2T_1R\] looks the same as
\[E_2BE_1T_2T_1R\] without the subscripts

Answer 32

the \(6!\) counts these as different, but in fact they are the same
\(6!\) overcounts by a factor of 4

Answer 33

thnx!

Answer 34

yw