Question

Sampath maintains her own library at home. she equally distributes her books on 8 shelves. if there are 27 books on a shelf, how many books does she have?​

Answer 1

Answer:

You can divide your task in steps, count the ways that you can conduct each step and then use the multiplication principle to count the total number of ways that you can conduct your task.

a) You can do it in two steps.

1.Step. Order your books. You can do it in 25! ways.

2.Step. Decide on how many books will be placed on each shelf. That is equivalent to writing on piece of paper 10 nonnegative (zeros are here allowed in c they are not!) integers x1,x2,…,x10 so that they add up to 25. For i=1,2,…,10, xi represents the number of books that will be placed on the i-th shelf. The total number of ways that you can conduct this step is equal to the number of solutions to the equation

x1+x2+…+x10=25,

with 0≤xi,i=1,2,…,10. This is a typical stars and bars problem with n=10 and k=25. Then (according to the Wikipedia page, Stars and Bars, Theorem two) you can do that in

(n+k−1n−1)=(10+25−110−1)=(349)

So in sum the task in (a) can be conducted in 25!×(349) ways.

c). Based on a) the answer is straightforward by allowing in Step 2. only positive values for the xi,i=1,2,…,10. The total number of ways that you can conduct this step is equal to the number of solutions to the equation

x1+x2+…+x10=25,

with 1≤xi,i=1,2,…,10. Then (according to the Wikipedia page, Stars and Bars, Theorem one) you can do it in (k−1n−1)=(25−110−1) ways.

So in sum the task in (c) can be conducted in 25!×(249) ways.

b) 25C25 is equal to 1 so, your approach here is not correct. Here you can think it as follows. Write on each book a number from 1 to 10 (this number indicates the shelf that it will be placed). So you can do it 10×10×…×10=1025 ways. (With steps as in the previous questions: 1.Step Choose a shelf for book Nr1. You can do it in 10 ways. 2.Step Choose a shelf for book Nr2. You can do it in 10 ways and so on until 25.Step Choose a shelf for book Nr25. You can do it in 10 ways. So by multiplication rule you can do it in 1025 ways