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 all five cards drawn are number cards and there are at least two cards of the same number

To find the probability that all five cards drawn from a pack of 52 playing cards are number cards and there are at least two cards of the same number, we need to follow these steps:

### 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: Ways to Draw All Number Cards
The number of ways to draw 5 cards from these 40 number cards is:
\[
\binom{40}{5} = \frac{40!}{5!(40-5)!} = 658,008
\]

### Step 4: Ways to Draw 5 Number Cards with No Repetition
The number of ways to draw 5 cards from the 40 number cards with no repetition (all different ranks) is calculated by selecting 5 different ranks from the 10 available, and then selecting one of the 4 suits for each:
\[
\binom{10}{5} \times 4^5 = 252 \times 1024 = 258,048
\]

### Step 5: Ways to Draw 5 Number Cards with At Least One Repetition
The number of ways to draw 5 number cards with at least one repetition is:
\[
\text{Total ways to draw all number cards} - \text{Ways to draw all different ranks}
\]
\[
658,008 - 258,048 = 399,960
\]

### Step 6: Probability Calculation
The probability is the ratio of the number of favorable outcomes (drawing 5 number cards with at least one repetition) to the total number of ways to draw 5 cards from 52:
\[
P(\text{5 number cards, at least one repetition}) = \frac{399,960}{2,598,960}
\]
\[
= \frac{399,960}{2,598,960} = \frac{9999}{64974} \approx 0.1538
\]

### Conclusion
The probability that all five cards drawn are number cards and there are at least two cards of the same number is approximately \(0.1538\) or \(15.38\%\).

Do you want more details or have any questions?

### Related Questions
1. What is the probability of drawing exactly two cards of the same number from the 40 number cards?
2. How many ways can you draw five cards from a deck such that all cards are face cards?
3. What is the probability that all five cards drawn are face cards?
4. How many ways can you draw 5 cards such that you get at least one Ace?
5. What is the probability that all five cards drawn are from the same suit?
6. How many combinations are there to draw five cards that include at least one King?
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 Jack and one Queen?

### Tip
When dealing with probabilities involving multiple conditions, break the problem into smaller parts and calculate the probabilities step by step.

Calculate the probability of drawing all five cards as number cards from a standard deck of 52 cards, with at least two cards of the same number. Learn the step-by-step solution using combinatorial principles and probability calculations.

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