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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.

 

Yunseol draws 5 cards from Ms. Q's deck. What is the probability that three cards have the same number?

 Nov 28, 2022
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If the problem is asking for exactly three:

There are 7*4 = 28 total cards
choose 3 colors -> 4c3

choose a number for the three colors to be -> 7c1

choose random other numbers to fill the rest of the hand -> 24c2

(4c3*7c1*24c2)/(28c5)

 

If asking for three or more: (basically > 3 cards that have the same number)

((ways of getting 3 card same) + (ways of getting 4 cards same)) / (ways to draw 5 cards)

= ((4c3*7c1*24c2)  + (4c4*7c1*24c1))/(24c5)

 Nov 28, 2022

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