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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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Yunseol can draw three cards with exactly two of the same color by following two scenarios:

 

Two cards from one color and one card from another color:

 

There are 4 choices of color for the two same-colored cards.

 

For each color choice, there are 7 options for the first card and 6 options for the second card (since two cards have the same color).

 

There are 4 choices of color for the different colored card.

 

Total ways for scenario 1 = 4 * (7 * 6) * 4 = 672

 

Three cards with one card from each of three different colors:

 

Yunseol can choose 3 colors out of 4 in 4C3 ways = 4.

 

For each color choice, there are 7 options for each card (since all three colors are different).

 

Total ways for scenario 2 = 4 * (7 * 7 * 7) = 1176

 

Adding the possibilities from both scenarios: 672 (scenario 1) + 1176 (scenario 2) = 1848

 

Therefore, Yunseol can draw 1848 three-card hands from Ms. Q's deck so that exactly two of the cards have the same color.

 Mar 28, 2024

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