how many ways are there to distribute seven different books among 10 people if no person is to receive more than one book?

Question

Subject of Combinations and Permutations

Answer 1

Am I reading too much into this problem ?  Does it matter which book each person gets or does it not matter so long as each person gets one book ?

Suppose there were 4 people and you want to distribute 3 books. In how many ways can you do this ?

 

Call the people 1, 2, 3, and 4, and call the books A, B, and C.

How many groups of 3 people can you form if there are 4 people ?

You can form 4 groups of 3 people. (4!/3!*1!)

They are:

123, 124, 134, and 234.

Now you want to distribute the books.

This is where my question comes in.

There are 6 different ways to distribute the 3 books to any one of the groups.

Person 1 chooses from 3 books, person 2 chooses from 2 books and person 3 gets the last book. (3*2*1=6)

Look at the first group of people, 123 and the books A,B, and C.

 

123

___

ABC

ACB

BAC

BCA

CAB

CBA

Person 1 gets book A, person 2 gets book B, and person 3 gets book C. or...

Person 1 gets book A, person 2 gets book C, and person 3 gets book B. or...

Person 1 gets book B, person 2 gets book A, and person 3 gets book C. or...

and so on...

There are 6 ways to distribute the 3 books among the group of 3 people. 

There are 4 groups of 3 people and 6 ways to distribute the books among each of the 4 groups.

Therefore there are 6*4 ways or 24 ways to distribute 3 books among 4 people.

In the original problem of distributing 7 books among 10 people...

How many groups of 7 people can you form from 10 people ?

(10!)/(7!)(3!)=120 groups. There are 120 groups of 7 people.

Now, again, does it matter which person gets which book ? Or is it sufficient that each person gets a book, no matter which book it is as long as each person gets one book ?

If it matters that each person can get 1 of 7 different books, then the number of ways to distribute 7 books among 10 people will increase greatly.

You have 120 groups of 7 people per group.

Take the first group of 7 people.

The first person chooses from 7 books, the second chooses from 6 books, the third chooses from 5 books and so on.

7*6*5*4*3*2*1=5040 ways to distribute 7 books among 7 people.

Therefore we have 120*5040=604,800 ways to distribute the seven books.

 

 

 

Answer 2

Hello, Brandi

 

We need to use the formula C(n,r)= [n!]/[(n-r)! (r!)

 

r= number of people who will received books.  In this case, 7 people. So, r=7

n= total number of people.  In this case the are 10 people.  So, n   will be 10

 

C(10,7)= [10!]/[(10-7)!(7!)

C(10,7)=[10*9*8*7!]/[(3!)(7!)]     The 7! will cancel each other out

             (10*9*8)/(3*2*1)

                  720/6

                    120

 

The are 120 different combination. 

 

D. Y. Taylor