Three cards are drawn at random from an ordinary pack of cards, find the chance that the selected combination will consists of one ace, one king and one queen.

Answer

To find the probability of drawing one Ace, one King, and one Queen from a standard deck of 52 cards, we can follow these steps:

### Step 1: Determine the Total Number of Ways to Choose 3 Cards from 52
The total number of ways to choose 3 cards from a deck of 52 cards is given by the combination formula \( nCr \), which is defined as:

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

For our case, \( n = 52 \) and \( r = 3 \):

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

### Step 2: Determine the Favorable Outcomes
Next, we need to find the number of ways to choose one Ace, one King, and one Queen.

- There are 4 Aces in the deck, so the number of ways to choose 1 Ace is \( \binom{4}{1} = 4 \).
- There are 4 Kings in the deck, so the number of ways to choose 1 King is \( \binom{4}{1} = 4 \).
- There are 4 Queens in the deck, so the number of ways to choose 1 Queen is \( \binom{4}{1} = 4 \).

Thus, the total number of favorable outcomes is:

\[ \text{Favorable outcomes} = \binom{4}{1} \times \binom{4}{1} \times \binom{4}{1} = 4 \times 4 \times 4 = 64 \]

### Step 3: Calculate the Probability
Now, we can calculate the probability of drawing one Ace, one King, and one Queen:

\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{64}{22100} \]

### Step 4: Simplify the Probability (if necessary)
We can simplify this fraction if possible. However, in this case, \( 64 \) and \( 22100 \) do not have common factors, so we leave it as:

\[ \text{Probability} = \frac{64}{22100} \]

### Final Answer
The probability that the selected combination will consist of one Ace, one King, and one Queen is:

\[ \frac{64}{22100} \]

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