2 cards are drawn simultaneously from a pack.
If x = no, of kings = 0, 1, 2, . Then P (x = 1) = ?

Options

A. `(32)/(229)`
B. `(32)/(227)`
C. `(32)/(221)`
D. `(32)/(219)`

Answer

To find the probability \( P(X = 1) \) where \( X \) is the number of kings drawn when two cards are drawn simultaneously from a standard deck of 52 cards, we can follow these steps:

### Step 1: Understand the total number of cards and kings
A standard deck has 52 cards, which includes 4 kings (one from each suit: hearts, diamonds, clubs, and spades) and 48 other cards.

### Step 2: Determine the total number of ways to draw 2 cards from 52
The total number of ways to choose 2 cards from 52 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Total ways} = \binom{52}{2} = \frac{52 \times 51}{2 \times 1} = 1326 \]

### Step 3: Determine the favorable outcomes for drawing 1 king
To have exactly 1 king in the 2 cards drawn, we need to choose:
- 1 king from the 4 kings
- 1 card from the remaining 48 non-king cards

The number of ways to choose 1 king from 4 is:
\[ \text{Ways to choose 1 king} = \binom{4}{1} = 4 \]

The number of ways to choose 1 card from the 48 non-kings is:
\[ \text{Ways to choose 1 non-king} = \binom{48}{1} = 48 \]

### Step 4: Calculate the total favorable outcomes
The total number of favorable outcomes for drawing 1 king and 1 non-king is:
\[ \text{Favorable outcomes} = \binom{4}{1} \times \binom{48}{1} = 4 \times 48 = 192 \]

### Step 5: Calculate the probability
Now, we can calculate the probability \( P(X = 1) \) using the formula:
\[ P(X = 1) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{192}{1326} \]

### Step 6: Simplify the probability
To simplify \( \frac{192}{1326} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 192 and 1326 is 6:
\[ P(X = 1) = \frac{192 \div 6}{1326 \div 6} = \frac{32}{221} \]

### Final Answer
Thus, the probability \( P(X = 1) \) is:
\[ \boxed{\frac{32}{221}} \]