Names of kids: Betty(5), Carol(5), Nick(4), and Joe(3)
Names of colours: green(5), pink(4), red(3), and yellow(6)

Clue #1: No person's name has the same number of letters as his or her favorite color.

From this we understand:
Betty - CANNOT LIKE - GREEN
Joe - CANNOT LIKE - RED
Nick - CANNOT LIKE - PINK
Carol - CANNOT LIKE - GREEN

Clue #2: Carol and the girl who likes pink are in the different grades.

From this we get that Carol and the girl who likes pink are seperate girls. There is only one other girl,Betty.She likes pink
So,Betty likes Pink

Clue #3: Red is the favorite color of one of the boys.

From our table above we know that,
Joe - CANNOT LIKE - RED
Nick - CANNOT LIKE - PINK

Therefore,Nick likes Red.

Now we know 2 of the favourite colours:
Betty - Pink and Nick - Red

Now we have 2 kids(Carol,Joe) and 2 colours(yellow,green) left

From our table above we know that,
Joe - CANNOT LIKE - RED
Carol - CANNOT LIKE - GREEN

Since there are only two colours left,Carol must like yellow.

Then Joe must like green.

To summarize,
Betty - pink
Carol - yellow
Nick - red
Joe - green

Hope this helps,good luck.