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

[davidtrinh]

(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]

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]

(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]

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