Four cards are drawn from a deck of playing cards. What is the probabi

Question

Question for the community : Please highlight the error in my solution to the below problem.

My solution:   \frac{{4C1*13C2*3C1*13C1*3C1*13C1}}{{52C4}}

Problem: Four cards are drawn from a deck of playing cards. What is the probability of getting exactly 2 cards of the same suit?

Official answer:   \frac{{12168}}{{20825}}

napolean92728 · Answer

We want  exactly 2 cards of the same suit , and the  other 2 cards to be of different suits (not matching each other or the pair). 
This means:
 
 One suit appears  twice  
 Two  other suits  appear  once each  
 The  fourth suit does not appear

Total number of ways to choose 4 cards from 52:
C(52, 4)

Step 1: Choose the suit that will appear twice
There are 4 suits → choose 1 of them → 4 ways

Step 2: Choose 2 cards from that suit
Each suit has 13 cards → choose 2 → C(13, 2)

Step 3: Choose 2 other suits (different from each other and the pair suit)
From remaining 3 suits → choose 2 → C(3, 2)

Step 4: From each of those 2 suits, choose 1 card
Each suit has 13 cards → 13 choices from each suit → 13 * 13 = 169

Total favorable outcomes =
4 * C(13, 2) * C(3, 2) * 13 * 13
 = 4 * 78 * 3 * 169
 = 4 * 78 * 507
 = 4 * 39546
 = 158184

Total possible outcomes = C(52, 4) = 270725

Final probability = 158184 / 270725

Dividing them both by 13 we get our final answer as : 12168 / 20825

Four cards are drawn from a deck of playing cards. What is the probabi

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