A teacher prepares 5 different books for 7 students.If each student can get one or no book from the teacher , find the number of ways of distributing the books among the students.
can u explain it to me detailly.i have thought about this question for serveral hours but i still cant get the answer
This is a tricky question.
The teacher prepares 5 books for 7 students, but did not say that all five books HAVE to be distributed. So my interpretation is that 0 to 5 books could have been distributed.
For the case of 0 book distributed, there is only one way: no one gets any book.
For 1 book distrbuted, there are 7 ways to distribute, one for each student.
For 2 books distributed, there are 7 ways for the first book, 6 ways for the second. The order of the books is not important because they are different. There are 7!/(7-2)! ways.
Continue until all 5 books are distributed for a total of
1+7+42+210+840+2520=3620 ways
but the correct ans is 2520,why?
As I indicated, this is a trick question.
I have assumed that not all five books need to be distributed because the question says the teacher "prepares" 5 books, and any student can take "one or no book". It did not say he "distributes" 5 books.
The answer assumes ALL five books are distributed, as you can verify with the last entry on my list, or 7!/2!
Sure! Let's break down the problem step by step.
First, let's consider the number of options for each student. Each student can either receive one book or no book. That means for each student, we have two choices.
Since there are 7 students, and for each student, we have two choices, we can calculate the total number of ways of distributing the books by multiplying the number of choices for each student together. This can be done using the concept of multiplication principle.
The multiplication principle states that if there are k independent events, and event i can occur in n_i ways, then the total number of different ways that all of the events can occur is the product of the number of ways that each event can occur.
In this case, we have 7 independent events (students) and each event (student) can occur in 2 ways (receive one book or no book). Therefore, the total number of ways of distributing the books among the students is 2 * 2 * 2 * 2 * 2 * 2 * 2, which can also be written as 2^7.
So, there are 2^7 = 128 different ways of distributing the books among the students.
I hope this explanation helps! Let me know if you have any further questions.