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

Options

A. `(2)/(219)`
B. `(1)/(221)`
C. `(3)/(209)`
D. `(1)/(209)`

Answer

To solve the problem of finding the probability \( P(X = 2) \), where \( X \) is the number of kings drawn from a pack of cards when 2 cards are drawn simultaneously, we can follow these steps:

### Step 1: Understand the Problem
We have a standard deck of 52 cards, which includes 4 kings (one from each suit). We want to find the probability of drawing exactly 2 kings when we draw 2 cards.

### Step 2: Identify Total Possible Outcomes
The total number of ways to choose 2 cards from a deck of 52 cards can be calculated using the combination formula \( nCr \), which is given by:

\[ nCr = \frac{n!}{r!(n-r)!} \]

For our case, we need to calculate \( 52C2 \):

\[ 52C2 = \frac{52!}{2!(52-2)!} = \frac{52 \times 51}{2 \times 1} = 1326 \]

### Step 3: Identify Favorable Outcomes
Next, we need to find the number of favorable outcomes where both cards drawn are kings. Since there are 4 kings in the deck, we can calculate the number of ways to choose 2 kings from these 4 kings:

\[ 4C2 = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 4: Calculate the Probability
The probability \( P(X = 2) \) is given by the ratio of the number of favorable outcomes to the total outcomes:

\[ P(X = 2) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{4C2}{52C2} = \frac{6}{1326} \]

### Step 5: Simplify the Probability
Now, we simplify \( \frac{6}{1326} \):

\[ P(X = 2) = \frac{6 \div 6}{1326 \div 6} = \frac{1}{221} \]

### Final Answer
Thus, the probability that both cards drawn are kings is:

\[ P(X = 2) = \frac{1}{221} \]

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