Calculating the Number of Ways to Draw 10 Cards with a Repetition

When faced with the problem of drawing a card from a deck of 52 cards, replacing it, and then drawing again, how many ways can 10 cards be drawn so that the 10th card is a repetition of a previous draw? This question delves into the intricate world of permutations, combinations, and probability. Let's break down the solution step-by-step, ensuring it aligns with Google's search and ranking standards.

Understanding the Problem

The core of this problem involves calculating the number of ways to draw 10 cards from a deck of 52 cards, with the final card being identical to one of the cards drawn in the previous 9 draws. This solution will be detailed, offering insights into the mathematical principles at play.

Step-by-Step Solution

1. Choosing the First 9 Cards

For each of the first 9 draws, there are 52 possible cards to choose from. This means the number of ways to draw the first 9 cards is:

[ 52^9 ]

2. Choosing the 10th Card

The 10th card must be one of the cards that has already been drawn in the first 9 draws. Since we can choose any one of the 9 previously drawn cards, there are 9 choices for this card.

3. Combining the Choices

The total number of ways to draw the 10 cards, ensuring that the 10th card is a repetition of a previous draw, can be calculated by multiplying the number of ways to choose the first 9 cards by the number of choices for the 10th card:

[ text{Total Ways} 52^9 times 9 ]

4. Final Calculation

The exact numerical value of this expression is:

[ 52^9 1125899906842624 ]

Now, multiplying by 9, we get:

[ text{Total Ways} 1125899906842624 times 9 10133089161583616 ]

Alternative Approach

Alternatively, we could find the total number of ways to draw 10 cards without any repetitions and then subtract that from the total possible ways to draw 10 cards with replacement. This approach simplifies the problem slightly:

1. Number of Ways to Draw 10 Cards Without Repetition

We calculate the number of ways to draw the first card (52 choices) and then each subsequent card must be different from the previous ones (51 choices for the second card, 50 for the third, and so on). Thus, the number of such ways is:

[ 52 times 51^9 ]

2. Total Number of Ways with Replacement

When drawing 10 cards with replacement, each draw has 52 possibilities, so:

[ 52^{10} ]

3. Number of Ways with at Least One Repetition

Subtracting the number of ways without repetitions from the total number of ways to draw 10 cards with replacement gives us:

[ 52^{10} - 52 times 51^9 ]

Conclusion

The total number of ways to draw 10 cards such that the 10th card is a repetition of a previous draw can be summarized as: