Three cards are drawn from a deck of 52 cards . What is the probability that
they are all spades ?

Answer

To find the probability that all three cards drawn from a deck of 52 cards are spades, we can follow these steps:

### Step 1: Determine the total number of outcomes
The total number of ways to draw 3 cards from a deck of 52 cards can be calculated using combinations. This is represented as \( \binom{52}{3} \).

\[ \text{Total outcomes} = \binom{52}{3} = \frac{52!}{3!(52-3)!} = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} = 22100 \]

### Step 2: Determine the number of favorable outcomes
The number of ways to draw 3 spades from the 13 spades available in the deck is also calculated using combinations. This is represented as \( \binom{13}{3} \).

\[ \text{Favorable outcomes} = \binom{13}{3} = \frac{13!}{3!(13-3)!} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286 \]

### Step 3: Calculate the probability
The probability that all three cards drawn are spades is given by the ratio of the number of favorable outcomes to the total number of outcomes.

\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{286}{22100} \]

### Step 4: Simplify the probability
To simplify \( \frac{286}{22100} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD).

Calculating the GCD, we find that:

\[ \frac{286 \div 2}{22100 \div 2} = \frac{143}{11050} \]

Thus, the simplified probability is:

\[ \text{Probability} = \frac{143}{11050} \]

### Final Answer
The probability that all three cards drawn are spades is \( \frac{143}{11050} \).

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