Find the probability that at least two of the chosen socks have the same color out of 18 socks- 2 pairs brown,3 pairs blue,4 pairs yellow. There are four brown, six blue and eight yellow socks in a drawer. Three socks are taken out of the drawer at random. Find the probability that at least two of the chosen socks have the same color.
Help is as usual, Hugely appreciated.

Option A:


*

*The total number of ways to select $3$ socks is $\binom{4+6+8}{3}=816$

*The number of ways to select $1$ sock of each color is $\binom41\cdot\binom61\cdot\binom81=192$

*Hence the probability for at least $2$ socks of the same color is $1-\frac{192}{816}$


Option B:


*

*The total number of ways to select $3$ socks is $\binom{4+6+8}{3}=816$

*The number of ways to select $3$ brown                 is $\binom43             =  4$

*The number of ways to select $3$ blue                  is $\binom63             = 20$

*The number of ways to select $3$ yellow                is $\binom83             = 56$

*The number of ways to select $2$ brown and $1$  blue   is $\binom42\cdot\binom61= 36$

*The number of ways to select $2$ brown and $1$  yellow is $\binom42\cdot\binom81= 48$

*The number of ways to select $2$ blue  and $1$  brown  is $\binom62\cdot\binom41= 60$

*The number of ways to select $2$ blue   and $1$ yellow is $\binom62\cdot\binom81=120$

*The number of ways to select $2$ yellow and $1$ brown  is $\binom82\cdot\binom41=112$

*The number of ways to select $2$ yellow and $1$ blue   is $\binom82\cdot\binom61=168$

*This adds up to $624$ ways for selecting at least $2$ socks of the same color

*Hence the probability for at least $2$ socks of the same color is $\frac{624}{816}$

