Two cards are selected at random from a deck of 52 playing cards. The probability that both the cards are greater than 2 but less than 9 is

Options

A. `46/221`
B. `63/221`
C. `81/221`
D. `93/221`

Answer

To solve the problem of finding the probability that both cards drawn from a deck of 52 playing cards are greater than 2 but less than 9, we can follow these steps:

### Step 1: Identify the total number of cards in the deck
A standard deck of playing cards contains 52 cards.

### Step 2: Determine the range of cards that are greater than 2 and less than 9
The cards that are greater than 2 and less than 9 are 3, 4, 5, 6, 7, and 8. This gives us a total of 6 cards in each suit (hearts, diamonds, clubs, spades).

### Step 3: Calculate the total number of favorable cards
Since there are 4 suits and each suit has 6 cards (3, 4, 5, 6, 7, 8), the total number of favorable cards is:
\[ 6 \text{ cards/suit} \times 4 \text{ suits} = 24 \text{ favorable cards} \]

### Step 4: Calculate the total number of ways to choose 2 cards from the deck
The total number of ways to choose 2 cards from 52 cards is given by the combination formula \( C(n, r) \), where \( n \) is the total number of items, and \( r \) is the number of items to choose. Thus:
\[ \text{Total outcomes} = C(52, 2) = \frac{52 \times 51}{2} = 1326 \]

### Step 5: Calculate the number of ways to choose 2 favorable cards
The number of ways to choose 2 cards from the 24 favorable cards is:
\[ \text{Favorable outcomes} = C(24, 2) = \frac{24 \times 23}{2} = 276 \]

### Step 6: Calculate the probability
The probability \( P \) that both cards drawn are greater than 2 and less than 9 is given by the formula:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{C(24, 2)}{C(52, 2)} = \frac{276}{1326} \]

### Step 7: Simplify the probability
Now we simplify \( \frac{276}{1326} \):
- Both numbers can be divided by 12:
\[ \frac{276 \div 12}{1326 \div 12} = \frac{23}{110.5} \]
However, we can also find the GCD to simplify further:
\[ \frac{46}{221} \]

Thus, the final probability that both cards are greater than 2 but less than 9 is:
\[ \frac{46}{221} \]

### Final Answer
The probability that both cards are greater than 2 but less than 9 is \( \frac{46}{221} \).

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