Find the number of ways in which from the set 1, 2, …, 49 you can choose six
different numbers so that at least two of them are consecutive.
I tried to solve it in this way:
Two consecutive numbers can be:
- 1,2 | 1,3 | 1,4 | … | 1,49
- 2,3 | 2,4 | 2,5 | … | 2,49
and so on.So total ways of picking two consecutive numbers will be 1+2+3+..+48 which is 1176. And number of ways of picking other 4 numbers will be 47 * 46 * 45 * 44. So total combination will be 1176 * 47 * 46 * 45 * 44. Am I right? (sorry for bad english)
Best Answer
The number of ways to choose 6 numbers will be $\binom{49}{6}$ and the number of ways to choose 6 non-consecutive numbers will be $\binom{44}{6}$ (refer to this).
Hence, the answer is $\binom{49}{6} - \binom{44}{6} = \boxed{6924764}$.