Question

# In a 5-card poker hand, how many possible hands are there where there are exactly 2...

In a 5-card poker hand, how many possible hands are there where there are exactly 2 cards of the same value plus 3 other cards with all different values none of which has the value 2. write down an explanation for the answer.

Deck Of Cards

A deck of 52 cards has 4 suits, Spades, Clubs, Diamonds and Hearts

There are 13 cards in each suit, from 2 till 10, the 3 face cards J, Q, K and finally Ace.

Therefore there are 4 cards of each type i.e 4 kings, 4 queens etc

2 cards should be the same (here 2 can also be considered), so we choose 1 card among 13 in 13C1 ways = 13

From the card chosen, we choose 2 cards in 4C2 = 6 ways

There are now 11 cards to choose from ( as we do not want to choose 2), and 3 cards can be chosen in 11C3 = 165

Then for each chosen card we choose 1 card out of 4 in 4C1 * 4C1 * 4C1 = 64 ways

Therefore Total Possible ways = 13 * 6 * 165 * 64 = 823,680

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