Five cards are drawn randomly from a pack of 52 playing cards (without jokers), where Ace, 2, 3, 4, 5, 6, 7, 8, 9 and 10 are all number cards.

(b) find the probability that only two different numbers are obtained from the cards drawn and at most three cards of the same number are drawn.

To find the probability that only two different numbers are obtained from the five cards drawn and at most three cards of the same number are drawn, we need to calculate the number of favorable outcomes and divide by the total possible outcomes.

### Step 1: Total Possible Outcomes
The total number of ways to draw 5 cards from 52 is given by the combination:
\[
\binom{52}{5} = \frac{52!}{5!(52-5)!} = 2,598,960
\]

### Step 2: Total Number Cards
There are 40 number cards in a deck (A, 2, 3, ..., 10 in each of the 4 suits).

### Step 3: Selecting Two Different Numbers
First, select 2 different numbers from the 10 available numbers (Ace to 10). This can be done in:
\[
\binom{10}{2} = \frac{10!}{2!(10-2)!} = 45
\]

### Step 4: Distributing 5 Cards Among 2 Chosen Numbers
We need to distribute 5 cards among the 2 chosen numbers such that at most 3 cards of each number are drawn. The valid distributions are (3, 2) or (2, 3):
- The number of ways to choose which number gets 3 cards and which gets 2 cards is 2 (since there are two ways to assign the numbers).

### Step 5: Selecting Cards from Each Number
For each valid distribution (3 cards of one number and 2 cards of another), we select the suits:
- Number of ways to choose 3 suits out of 4 for the first number: \(\binom{4}{3} = 4\)
- Number of ways to choose 2 suits out of 4 for the second number: \(\binom{4}{2} = 6\)

Thus, the total number of ways for each distribution is:
\[
\binom{4}{3} \times \binom{4}{2} = 4 \times 6 = 24
\]

Since there are 2 ways to assign the numbers, we multiply by 2:
\[
2 \times 24 = 48
\]

### Step 6: Total Number of Favorable Outcomes
Multiply the number of ways to select 2 numbers by the number of ways to distribute the cards:
\[
45 \times 48 = 2,160
\]

### Step 7: Probability Calculation
The probability is the ratio of the number of favorable outcomes to the total number of ways to draw 5 cards from 52:
\[
P(\text{2 different numbers, at most 3 of each}) = \frac{2,160}{2,598,960}
\]
\[
= \frac{2,160}{2,598,960} \approx 0.000831
\]

### Conclusion
The probability that only two different numbers are obtained from the cards drawn and at most three cards of the same number are drawn is approximately \(0.000831\) or \(0.0831\%\).

Do you want more details or have any questions?

### Related Questions
1. What is the probability of drawing exactly three cards of one number and two of another?
2. How many ways can you draw five cards from a deck such that they are all of the same suit?
3. What is the probability of drawing a full house (three of one number and two of another)?
4. How many ways can you draw 5 cards that include at least one face card?
5. What is the probability that all five cards drawn are from the same number?
6. How many combinations are there to draw five cards that include at least one card from each suit?
7. What is the probability of drawing four cards of the same rank and one card of a different rank?
8. How many ways are there to draw five cards that include at least one Ace?

### Tip
Always clearly define the conditions of the problem and ensure each step logically follows the previous one when solving probability problems.

Explore the probability of drawing exactly two different numbers from five randomly drawn cards from a standard 52-card deck. Learn how to calculate the probability using combinatorial methods and understand the conditions required for this scenario. Suitable for high school students studying probability and combinatorics.

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