Question:
Four cards are drawn at random from a pack of 52 playing cards. The probability of getting all four cards of the same suit is
Four cards are drawn at random from a pack of 52 playing cards. The probability of getting all four cards of the same suit is
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When finding probability of all cards from the same suit, calculate the combinations for one suit and multiply by the number of suits, then divide by total combinations of 4 cards from 52.
Updated On: Jul 18, 2026
- \(\frac{13}{270725}\)
- \(\frac{91}{190}\)
- \(\frac{178}{20825}\)
- \(\frac{44}{4165}\)
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The Correct Option is D
Solution and Explanation
Step 1: Total ways to draw 4 cards.
Total ways \(= \binom{52}{4} = \frac{52 \cdot 51 \cdot 50 \cdot 49}{4 \cdot 3 \cdot 2 \cdot 1} = 270725\).
Step 2: Ways to draw 4 cards from the same suit.
Each suit has 13 cards, so \(\binom{13}{4} = 715\) ways for one suit.
There are 4 suits, so total ways \(= 4 \cdot 715 = 2860\).
Step 3: Probability formula.
\[ P(\text{all same suit}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]
Step 4: Apply numbers.
\[ P(\text{all same suit}) = \frac{2860}{270725} \]
Step 5: Simplify the fraction.
\[ \frac{2860}{270725} = \frac{44}{4165} \]
Step 6: Final conclusion.
Hence, the probability that all four cards are of the same suit is \[ \boxed{\frac{44}{4165}} \]
Total ways \(= \binom{52}{4} = \frac{52 \cdot 51 \cdot 50 \cdot 49}{4 \cdot 3 \cdot 2 \cdot 1} = 270725\).
Step 2: Ways to draw 4 cards from the same suit.
Each suit has 13 cards, so \(\binom{13}{4} = 715\) ways for one suit.
There are 4 suits, so total ways \(= 4 \cdot 715 = 2860\).
Step 3: Probability formula.
\[ P(\text{all same suit}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]
Step 4: Apply numbers.
\[ P(\text{all same suit}) = \frac{2860}{270725} \]
Step 5: Simplify the fraction.
\[ \frac{2860}{270725} = \frac{44}{4165} \]
Step 6: Final conclusion.
Hence, the probability that all four cards are of the same suit is \[ \boxed{\frac{44}{4165}} \]
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