Question 1060190
It is easy to get mixed up and make a mistake in the cslculations, so I made a diagram.
All the possibilities can be represented in the diagram below.
The numbers next to the left edge represent the number of ways to pick the first sock in each color.
If the first sock was put back before picking a second sock,
The diagram would be a perfect square, divided into even rows and columns.
Since you do not replace, the diagram is a rectangle with slightly mismatched columns.
The numbers along the horizontal lines show the width of each column.
{{{drawing(500,500,-3,17,-1,19,
rectangle(0,0,16,17),
rectangle(0,0,6,7),rectangle(10,0,16,7),
rectangle(6,7,9,11),rectangle(9,7,16,11),
rectangle(0,11,5,17),rectangle(5,11,9,17),
locate(-2.8,17.5,first),locate(-2.8,17,sock),
locate(-2.8,4,green),locate(-2.8,9,blue),
locate(-2.8,14,red),locate(2,18.8,red),
locate(6.5,18.8,blue),locate(12,18.8,green),
locate(0.1,4,7),locate(0.1,9,4),locate(0.1,14,6),
locate(2.5,17,5),locate(7,17,4),locate(12.5,17,7),
locate(7.5,11,3),locate(3,11,6),
locate(8,7,4),locate(13,7,6),
locate(-1.5,19,second),locate(-1.5,18.5,sock)
)}}} 
The number of ways to get any color sequence for the socks picked
is the "area" of the corresponding rectangle
(the product of its width and height numbers).
On paper, I wrote the "area" inside each of the small rectangles,
and I also added the areas of each row and column.
(The numbers are repeated).
I did not write that much in the diagram, because it takes a lot of time to doing it through the website.
For example, there are 4 ways to get a blue sock in the first draw,
and after that there are 3 blue socks left.
That is shown as the height and width of the middle rectangle.
Their product, {{{4*3=13}}} is the number of ways to pick 2 blue socks.
You could count them if you had a way to tell one blue sock apart from another.
Similarly,the number of ways to pick a red sock first, and then a blue one is
{{{6*4=24}}} , which is represented by the 6 by 4 rectangle in the middle of the top row.
The case of a red second sock is represented by the (uneven) left column.
The top left rectangle represents the {{{6*5=30}}} ways to get two red socks.
The rectangle below that shows the {{{4*6=24}}} ways to get a blue sock first, and then a red one.
The rectangle beow that represents the {{{7*6=42}}} ways to get a green sock first, and then a red one.
The total ways of ending with a red sock for the second draw is
{{{30+24+42=96}}} , and only in {{{30}}} of those cases were both socks red.
That means, the answer to part a) is {{{30/96=5/16}}} .
For part c), the whole big rectangle represents the {{{16*17=272}}} ways to get two socks in any color sequence.
Since there were {{{96}}} ways to have get a red sock in the second draw,
there are {{{272-96=176}}} ways to get a second sock that is not red,
represented by the two left columns.
The part of the blue middle row  in the two left (non-red) columns represents
the {{{4*3+4*7}}} or {{{4*10=40}}} ways to get a first blue sock,
without getting a second red sock.
Those {{{40}}} ways out of the {{{176}}} ways of avoiding a second red sock are
{{{40/176=5/22}}} .