Four Cards Are Drawn At Random From A Full Pack. What Is The Probability That They Belong To 4 Different Suits?

The 4 cards may be assumed to have been drawn one by one without replacement. The first card can be drawn in 52 ways, the second card in 51 ways, third in 50 and the fourth in 49 ways. So, the total number of ways in which the card can be drawn is 52 X 51 X 50 X 49.

In order to find out the number of cases favourable to the event 'one card drawn from each suit', we see that the first drawn card may belong to any suit, and hence may be chosen in 52 ways. There now remains 51 cards of which 12 belong to the same suit as the first card and 39 belong to other suits. Since the 4 cards are to belong to different suits, the second card should come from the 39. Thus the second card may be drawn in 39 ways; similarly the third card in 26 ways and the fourth card in 13 ways. The number of favourable cases is therefore 52 X 39 X 26 X 13. The required probability is

(52 X 39 X 26 X 13) / (52 X 51 X 50 X 49) = 2197 / 20825

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