In Ms. Q's deck of cards, every card is one of four colors (red, green, blue, and yellow), and is labeled with one of seven numbers ($1,$ $2,$ $3,$ $4,$ $5,$ $6,$ and $7$). Among all the cards of each color, there is exactly one card labeled with each number. The cards in Ms. Q's deck are shown below.
Yunseol draws five cards from Ms. Q's deck. What is the probability that exactly three cards have the same number?
There are $\binom{28}{5}=32760$ total combinations of five cards that Yunseol could draw.
If exactly three cards have the same number, we need to choose which number this is, and then choose which color it is.
If one number has three cards, there are $\binom73=35$ ways to choose which three colors it is.
If the remaining two cards have different numbers, then the first remaining card has $4$ choices for color, and the second remaining card has $3$ choices for color. This gives $35\times 4\times 3=420$ ways to choose the five cards.
If the remaining two cards have the same number, then there are $6$ ways to choose which number it is. Then the first remaining card has $4$ choices for color, and the second remaining card has $3$ choices for color. This gives $35\times 6\times 4\times 3=2520$ ways to choose the five cards.
Thus the probability is $\dfrac{420 + 2520}{32760} = \dfrac{2940}{32760} = \boxed{\dfrac{1}{11}}$.