# tricky problem: 7 books on shelf; find number of ways to pick, arrange 4

#### davidtrinh

##### New member
(ex) A student has 7 different textbooks. How many different ways of selecting and arranging 4 books on a shelf are there?

my attempt

I do seven choose four to get 7C4 = 35. But, the answer key says 210.

How do you get the answer? Thanks a lot.

#### MarkFL

##### Super Moderator
Staff member
You did well to begin with:

$$\displaystyle \displaystyle {7 \choose 4}=35$$

Now, you need to observe that there are 4! ways to order the 4 books chosen...so the final answer is?

Another way to approach this is to think of the fact that there are 7 choices for the first position, 6 for the second, 5 for the third and 4 for the fourth:

$$\displaystyle N=7\cdot6\cdot5\cdot4$$

But...this returns a number 4 times as large as that you've cited as being the answer.

#### Dr.Peterson

##### Elite Member
(ex) A student has 7 different textbooks. How many different ways of selecting and arranging 4 books on a shelf are there?

my attempt

I do seven choose four to get 7C4 = 35. But, the answer key says 210.

How do you get the answer? Thanks a lot.
Since you are arranging, not just selecting, the correct answer is not 7C4 but 7P4 = 7!/3! = 840.

It looks like they may have accidentally done 7!/4!, which is easy to do if you aren't thinking.

#### davidtrinh

##### New member
Thanks a lot, MarkFL and Dr.Peterson. So, the correct answer is 840. I believe the book's answer is wrong.