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Fabian has a deck of 25 cards. Each card has an integer between 1 and 5, inclusive, printed on it in one of five colors: red, orange, green, blue, or violet. Each number-color combination appears on exactly one card in the deck. Fabian draws four cards at random, without replacement, from the deck. What is the probability that exactly two different numbers and exactly three different colors appear on his four cards? Express your answer as a common fraction.

 Jul 16, 2022
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Hi RainbowSquirrel53, 

 

Fabian has a deck of 25 cards.

Each card has an integer between 1 and 5, inclusive, printed on it in one of

five colors: red, orange, green, blue, or violet.

Each number-color combination appears on exactly one card in the deck.

 

Fabian draws four cards at random, without replacement, from the deck.

What is the probability that exactly two different numbers and exactly three different colors appear on his four cards? Express your answer as a common fraction.

----------------

 

I'm not certain but here is my logic.

Say tthe numbers are 1,2 and 3 and the colours are red and orange.  There are only 6 such cards so it is impossible to chose only 1 number or one colour shen chosing 4 of them.  That makes the problem a bit more simple than it might otherwise have been.

 

there are   5C2 ways to chose 2 numbers and 5C3 ways to chose 3 colours then 6C4 ways to chose the exact cards

that is    5C2 * 5C3 * 6C4 = 10*10*15 = 1500 possible favourable combinations

 

There are   25C4 = 12650 possible combinations

 

1500 / 12650 = 30/253    which is about 12% chance

 Jul 17, 2022
edited by Melody  Jul 17, 2022

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