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# Ms. Q's deck of cards...

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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.

How many 3 card hands can Yunseol draw from Ms. Q's deck, so that exactly two of the cards have the same color?

Mar 28, 2024

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There are two cases to consider:

Case 1: Two cards are from the same color and one card from a different color.

Choose one of the four colors for the two matching cards: There are 4 ways to do this.

Choose two out of the seven cards of that chosen color: There are ⁷C₂ ways to do this (combination, order doesn't matter).

Choose one card from each of the remaining three colors: There are 3 * 3 * 3 = 27 ways to do this (since there are 3 colors left and 3 cards of each color).

Case 2: Three cards are from different colors.

Choose one card from each of the four colors: There are 4 ways to do this.

Adding the number of ways for each case:

Total hands = (Ways in Case 1) + (Ways in Case 2)

Total hands = (4 * ⁷C₂ * 27) + 4

We can calculate ⁷C₂ using the combination formula: ⁷C₂ = ⁷! / (2! * (⁷ - 2)!) = 21

Total hands = (4 * 21 * 27) + 4

Total hands = 2520 + 4

Total hands = 2524​