From a pack of 52 cards, 3 cards are drawn together at random , what is the probability of all the cards are king ?

Options

A. `(1)/(5225)`
B. `(1)/(5525)`
C. `5525`
D. `(1)/(525)`

Explanation

To find the probability that all three cards drawn from a pack of 52 cards are kings, 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 can be calculated using the combination formula:

\[ \text{Total ways} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Where:
- \( n \) is the total number of items (52 cards),
- \( r \) is the number of items to choose (3 cards).

So, we calculate:

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

### Step 2: Determine the number of ways to draw 3 kings from the 4 kings available.
Since there are 4 kings in a standard deck of cards, the number of ways to choose 3 kings from these 4 can also be calculated using the combination formula:

\[ \text{Ways to choose 3 kings} = \binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4}{1} = 4 \]

### Step 3: Calculate the probability.
The probability \( P \) of drawing 3 kings can be calculated as the ratio of the number of favorable outcomes to the total number of outcomes:

\[ P(\text{3 kings}) = \frac{\text{Ways to choose 3 kings}}{\text{Total ways to choose 3 cards}} = \frac{4}{22100} \]

### Step 4: Simplify the probability.
To express this probability in a simpler form, we can calculate:

\[ P(\text{3 kings}) = \frac{4}{22100} \approx 0.000181 \]

### Final Answer:
Thus, the probability of drawing 3 kings from a pack of 52 cards is:

\[ P(\text{3 kings}) = \frac{4}{22100} \text{ or approximately } 0.000181 \]

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