Question

Combinatorics:
What are the total number of ways that a string of length n can be formed if it consists of letters and must contain an even number of x's.

Answer 1

can you explain a bit?

Answer 2

string is of any length, contains letters, and must contain an even number of x's, meaning 0 x's is still okay

Answer 3

how many possible strings of length n can be formed given those rules

Answer 4

looks pretty difficult

Answer 5

the professor asks for a recursive definition, not exact formula, so (n-1) applies if we have the base cases.

Answer 6

if n is even then,

Answer 7

I think it has to do with adding an x to a bad string and adding another letter to a good string, therein lies a recursive definition.

Answer 8

i guess it's better to try it without x's first.

Answer 9

if n<26,
26Cn
what if n>26 ??

Answer 10

oh ... even this is incorrect. we are able to choose any letter any number of times right!!

Answer 11

yes

Answer 12

for x < 26,
\[ \sum_{n=1}^{26} \binom{26}{n}\binom{x}{n}\]

Answer 13

seems this is incorrect too.

Answer 14

number of bad strings are those strings with an odd number of x's

Answer 15

very puzzling

Answer 16

this should work out for string of length less than 26
\sum_{n=1}^{x} \binom{26}{n}\binom{x}{n}

Answer 17

\[ \sum_{n=1}^{x} \binom{26}{n}\binom{x}{n}\]

Answer 18

it should be easier than this, as the definition needs to be recursive, and base cases are easy to determine, maybe I'll chart possibilities up to 5 and see what it looks like