+0 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.
Note: Every line is a new color
1 2 3 4 5 6 7
1 2 3 4 5 6 7
1 2 3 4 5 6 7
1 2 3 4 5 6 7
How many $3$ card hands can Yunseol draw from Ms. Q's deck, so that exactly two of the cards have the same color?
+193 There are two cases to consider:
Case 1: Two cards of the same color come from the same pile.
Choose the color of the two cards in 4 ways (red, green, blue, or yellow).
Choose two different numbers from the chosen color in 7C2 ways (7 ways to choose the first number, then 6 ways to choose the second number given the first one has already been chosen).
Choose the third card, which can be any of the remaining 21 cards in 21 ways.
The total number of hands in this case is 4 * 7C2 * 21 = 504.
Case 2: Two cards of the same color come from different piles.
Choose two different colors in 6C2 ways (6 ways to choose the first color, then 5 ways to choose the second color given the first one has already been chosen).
Choose one number from each of the chosen colors in 7 * 7 = 49 ways.
Choose the third card, which can be any of the remaining 19 cards in 19 ways.
The total number of hands in this case is 6C2 * 49 * 19 = 1716.
Total number of hands:
Adding the number of hands from both cases, we get 504 + 1716 = 2220.
Therefore, Yunseol can draw 2220 three-card hands with exactly two cards of the same color.
