Question

Let X and Y be two languges over the alphabet {0, 1} such that ,

$X=|w|w\in {0,1}^3;n_0\left(w\right)=n_2\left(w\right)$

$Y=|w|w\in {0,1}^{49};n_0\left(w\right)=n_1\left(w\right)$

If X and Y denote the cardinalities of X of Y respectively, then the value of X + Y is equal to ____.

Answer 1

Indirectly the question is, " Number of bit strings of length n having equal number of 0's and 1's , which will be equal to "$C_{n/2}$ if n is even, 0 if in is odd (as it's not possible to set equal 0's and 1's in case of an odd length bit string).

Let's find |X| first.
X consist of binary string of length 8 having equal 0's and 1's.
Since 8 is even , hence |X| = 8C4 = 70

But |Y| will be 0, because 49 is odd, and its not possible to have an odd length string with equal number of 0's and 1's.
Hence, |Y| = 0.

Therefore, their sum is equal to 70.