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Name Morkredd – ODE
Publisher talmar
Format File
Rating 4.45 / 5 ( 1832 votes )
Update (10 days ago)

 

 

 

 

 

 

Q:

What is the number of ways of placing $10$ cards with the same rank?

The problem is: I have $10$ cards with ranks $1,2,\ldots,10$. In how many ways can I arrange them?

My attempt:
I did something like this:
$$\frac{10!}{3!\times 7!}=\frac{10!}{3!(4!)^2}=\frac{10!}{9!\times 8}$$
Is that the answer?

A:

Your solution is almost correct. You forgot the factor of $1/2!$:
$$\frac{10!}{3!(4!)^2} = \frac{10!}{9!\times 8} \times \frac{4!}{2!} = \frac{10!}{4!\times8\times 7} = \frac{10!}{16\times 7} = \frac{10!}{8\times 5\times 6} = \frac{10!}{8\times 6\times 6} = \frac{10!}{96} = \frac{10!}{72} = \frac{10!}{24}$$

A:

Approach 1: How many ways can you arrange the cards?
The first card can be in any of the positions {1, 2,…, 10}. If that’s the only card, there are $10$ ways to do it. Now, if you only have 3 cards, that makes it easy to calculate that there are $1$ way to arrange them since you can place the card anywhere. That’s $10$ ways to arrange the first card. The same goes for 4, 5,…, 9 cards.
Approach 2: How many ways can you arrange the cards?
For the second card, you have $9$ options to choose from. Again, the only card that has no way to be chosen, because there are only 9 cards, is the first card. So, you have:
$$\frac{9!}{1!\cdot 8!} = \frac{9!}{8! \times 7} = \frac{9!}{8!\times 6} = \frac{9!}{8!\times 6} = \frac{9!}{8

 

Name Morkredd – ODE
Publisher talmar
Format File
Rating 4.45 / 5 ( 1832 votes )
Update (10 days ago)

 

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    at 11.25.04 GMT

    11.25.04 GMT

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