Odd & Even Combinations If you have a range of numbers from 1-49 with 6 numbers to choose from, how many combinations are there containing all odd, all even, only 1 odd, and only 1 even number?
 A: All 6 odd combinations:
$$\ _{25}C_6 / _{49}C_6 = 177100/13983816 $$
All 6 even combinations:
$$\ _{24}C_6 / _{49}C_6 = 134596/13983816 $$
Only 1/6 odd combinations:
$$\ _{25}C_5 * 24 / _{49}C_6 = 1275120/13983816 $$
Only 1/6 even combinations:
$$\ _{24}C_5 * 25 / _{49}C_6 = 1062600/13983816 $$
I use choose rather than pick because you do not want repeats.
A: There are $25$ odd and $24$ even numbers in the range $1 - 49$.  
The number of selections of six numbers from the set $\{1, 2, 3, \ldots, 49\}$ that contain exactly $k$ odd numbers and $6 - k$ even numbers is 
$$\binom{25}{k}\binom{24}{6 - k}$$

  Therefore, the number of selections that contain just odd numbers is $$\binom{25}{6}\binom{24}{0} = \binom{25}{6}$$ The number of selections that contain just even numbers is $$\binom{25}{0}\binom{24}{6} = \binom{24}{6}$$  Selections that contain exactly one odd number must contain five even numbers.  The number of such selections is $$\binom{25}{1}\binom{24}{5}$$  Selections that contain exactly one even number must contain five odd numbers.  The number of such selections is $$\binom{25}{5}\binom{24}{1}$$

