Three cards are drawn at a time at random from a well shuffled [ack of 52 cards . Find the probability that all the three cards have same number .

Answer

To find the probability that all three cards drawn from a well-shuffled deck of 52 cards have the same number, we can follow these steps:

### Step 1: Determine the total number of ways to draw 3 cards from 52 cards.
The total number of ways to choose 3 cards from 52 is given by the combination formula \( \binom{n}{r} \), which is calculated as:

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

### Step 2: Determine the number of favorable outcomes where all three cards have the same number.
In a standard deck of cards, there are 13 different ranks (Ace, 2, 3, ..., 10, Jack, Queen, King). For each rank, there are exactly 4 cards (one for each suit). To have all three cards of the same rank, we can choose any one of the 13 ranks and then choose 3 out of the 4 cards of that rank.

The number of ways to choose 3 cards from 4 cards of the same rank is given by:

\[ \text{Favorable outcomes} = \binom{4}{3} \times 13 = 4 \times 13 = 52 \]

### Step 3: Calculate the probability.
The probability \( P \) that all three cards drawn have the same number is given by the ratio of the number of favorable outcomes to the total number of outcomes:

\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{52}{22100} \]

### Step 4: Simplify the probability.
To simplify \( \frac{52}{22100} \):

\[ P = \frac{52 \div 52}{22100 \div 52} = \frac{1}{425} \]

### Final Answer:
The probability that all three cards drawn have the same number is \( \frac{1}{425} \).

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