Question

How many seven-letter words contain at least two X's? Hint: The Bad ones
are those with no X's and those with exactly one X. Think carefully about
counting the number of words with exactly one X.

Answer 1

@tim93 Find the no. of words with no X's?

Answer 2

= all possible combinations
= all possible combinations with no X
= all possible combinations with 1 X

Any thoughts for any of them ?

Answer 3

@tim93
yeah try this after you get up:D
Night:)

Answer 4

All possible combinations = 26^7
All possible combinations with no x = 25^7
All possible combinations with only one x = ?????? getting stuck here

Answer 5

word means any string of 7 letters?

Answer 6

yes

Answer 7

no \(x\) them would be \(25^7\) i guess

Answer 8

maybe only one x would be 26^6 + 26^7??

Answer 9

one \(x\) would be \(7\times 25^6\) i think
here is my reasoning, see if it make sense
x in the first spot, leaving 6 places and 25 choices or
x in the 2nd spot, leaving 6 places and 25 choices or
...

Answer 10

wait,

Answer 11

26 * 6^25 for wods with only one x?

Answer 12

With only one x would be 26^6 to choose the other 6 letters, and then multiply that by 7 because the x could be placed 7 different spots.

Answer 13

you have the base and exponent backwards i think

Answer 14

so the final equation would be ...
26^7 - 25^7 - (26 * 25^6) ???
sound right?

Answer 15

x, __, __, __, __, __, __ 6 slots in which each has 25 choices

Answer 16

ok \(26^7\) is the total with no restriction right ?

Answer 17

with the restriction that there is no x it is \(25^7\)

Answer 18

and with the restriction that there is one x i still believe by what i wrote above that it is \(7\times 25^7\)

Answer 19

maybe i am wrong, track record not so good today

Answer 20

26^7 - 25^7 - (7* 25^7)

Answer 21

oh yes i am wrong
\[7\times 25^6\]

Answer 22

Tim, to create a word with only 1 x, you must choose the other 6 letters. For each of those, you have 25 options, so to choose those letters, you get this many possibilities
25^6
Now, you've chosen all of the other 6 letters, but you can still get different choices by placing the x in different spots. For example,
xaaaaaa
axaaaaa
aaxaaaa
So, that means there are 7 different spots to place the x, therefore the total number of possibilities is
7*25^6

Answer 23

\[\sum_{x=2}^{7}{7\choose x}25^{7-x}\]
\[=26^7-\sum_{x=0}^{1}{7\choose x}25^{7-x}\]

Answer 24

zarkon gives direct computation

Answer 25

26^7 - 25^7 - (7 * 25^6)

Answer 26

Good.

Answer 27

what smoothmath said

Answer 28

thanks guys