From a pack of 52 cards , 3 cards are drawn at random . Find the probability of drawing exactly two aces .

Answer

To find the probability of drawing exactly two aces when three cards are drawn from a standard deck of 52 cards, 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 a deck of 52 cards can be calculated using the combination formula:

\[ \text{Total ways} = \binom{52}{3} \]

Calculating this:

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

### Step 2: Determine the number of ways to choose 2 aces from the 4 aces
Next, we need to find the number of ways to choose 2 aces from the 4 available aces in the deck:

\[ \text{Ways to choose 2 aces} = \binom{4}{2} \]

Calculating this:

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

### Step 3: Determine the number of ways to choose 1 non-ace from the remaining cards
After selecting 2 aces, we need to choose 1 card from the remaining 48 cards (since there are 52 cards in total and 4 are aces):

\[ \text{Ways to choose 1 non-ace} = \binom{48}{1} \]

Calculating this:

\[ \binom{48}{1} = 48 \]

### Step 4: Calculate the total number of favorable outcomes
The total number of favorable outcomes for drawing exactly 2 aces and 1 non-ace is the product of the number of ways to choose the aces and the non-aces:

\[ \text{Total favorable outcomes} = \binom{4}{2} \times \binom{48}{1} = 6 \times 48 = 288 \]

### Step 5: Calculate the probability
Now, we can find the probability of drawing exactly 2 aces by dividing the total number of favorable outcomes by the total number of ways to draw 3 cards:

\[ \text{Probability} = \frac{\text{Total favorable outcomes}}{\text{Total ways}} = \frac{288}{22100} \]

### Step 6: Simplify the probability
To simplify the fraction:

\[ \frac{288}{22100} \approx 0.01306 \]

Thus, the probability of drawing exactly 2 aces when 3 cards are drawn from a pack of 52 cards is approximately \(0.01306\) or \(\frac{72}{5525}\) when simplified.

### Final Answer:
\[ \text{Probability} = \frac{72}{5525} \]