1. ## subsets question

Hey iv been stuck on this question for a while now and am convinced its just because im missing something obvious so was wondering if anyone can give me some help.

What proportion of 6-element subsets of {1, 2, . . . , 49} contain at least two consecutive integers? What proportion of 7-element subsets of {1, 2, . . . , 49} contain at least two con-secutive integers?

thanks

2. Originally Posted by CHAYNES
What proportion of 6-element subsets of {1, 2, . . . , 49} contain at least two consecutive integers? What proportion of 7-element subsets of {1, 2, . . . , 49} contain at least two consecutive integers?
How many bit-strings of length 49 contain exactly 6 ones but no two ones are consecutive?
The answer is $\binom{44}{6}$. The 6 ones are separated by at least one zero.
This the number of 6-element subsets of {1, 2, . . ., 49} containing no consecutive integers.
The answer to the question is $2^{49}-\binom{44}{6}$.