Four cards are drawn from a well shuffled deck of 52 cards. Number of ways to draw four cards of same denomination `=alpha` and number of ways to draw four cards of different denomination `=beta`. Which of following is correct?

Options

A. a. `(beta)/(alpha)` is divisible by 13.
B. b. `beta+alpha` is divisible by 13
C. c. `beta-alpha` is divisible by 13
D. d. `beta` is divisible by `(13)^(2)`

Explanation

To solve the problem, we need to calculate the number of ways to draw four cards of the same denomination (denote this as α) and the number of ways to draw four cards of different denominations (denote this as β).

### Step 1: Calculate α (Four cards of the same denomination)

1. **Understanding the problem**: There are 13 different denominations in a deck of cards (Ace, 2, 3, ..., 10, Jack, Queen, King). Each denomination has 4 cards.
2. **Choosing the denomination**: To draw four cards of the same denomination, we can choose any one of the 13 denominations.
3. **Calculating α**: Since there is only one way to choose all four cards from a single denomination, the total number of ways to draw four cards of the same denomination is simply the number of denominations:
\[ \alpha = 13 \]

### Step 2: Calculate β (Four cards of different denominations)

1. **Choosing the denominations**: We need to select 4 different denominations from the 13 available. The number of ways to choose 4 denominations from 13 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Ways to choose 4 denominations} = \binom{13}{4} \]
2. **Choosing one card from each chosen denomination**: For each of the 4 chosen denominations, we can choose 1 card from the 4 available cards. Thus, for each of the 4 denominations, there are 4 choices:
\[ \text{Ways to choose 1 card from each denomination} = 4^4 \]
3. **Calculating β**: Therefore, the total number of ways to draw four cards of different denominations is:
\[ \beta = \binom{13}{4} \times 4^4 \]

### Step 3: Calculate the values

1. **Calculate \( \binom{13}{4} \)**:
\[ \binom{13}{4} = \frac{13!}{4!(13-4)!} = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = 715 \]
2. **Calculate \( 4^4 \)**:
\[ 4^4 = 256 \]
3. **Now, calculate β**:
\[ \beta = 715 \times 256 = 183040 \]

### Summary of Results
- \( \alpha = 13 \)
- \( \beta = 183040 \)

### Final Answer
The values of α and β are:
- \( \alpha = 13 \)
- \( \beta = 183040 \)