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Chapter 1: Problem 14

In how many ways can a teacher distribute 12 different science books among 16 students if (a) no student gets more than one book? (b) the oldest student gets two books but no other student gets more than one book?

Short Answer

Expert verified
For part (a), the teacher can distribute the books in 43546182400 different ways. For part (b), the teacher can distribute the books in 8789181600000 different ways.

Step by step solution

01

Calculating permutation for part (a)

In part (a), we are trying to distribute 12 different books to 16 different students such that no student gets more than one book. This is a permutation problem, treating the books as spaces to assign to the students and since a book can't be given to more than one student. Using the formula for permutation without repetition, \(nPn\), we have \(16P12 = \frac{16!}{(16-12)!}\).
02

Evaluating the permutation for part (a)

Using the formula from step 1, we substitute the values into the equation to compute the result. Factorial can be done manually (multiplying each integer from 1 to the desired number) or using a calculator. Evaluating \(16P12\), we find that the teacher can distribute the books in \(43546182400\) ways such that no student receives more than one book.
03

Calculating permutation for part (b)

In part (b), it's specified that the oldest student gets two books and no other student gets more than one book. So, after giving two books to the oldest student, 10 books will remain to distribute among 15 students. Here we will calculate permutation twice because we have two different conditions. First, we'll calculate the ways to select 2 books out of 12 to give to the oldest student, which is \(12P2 = \frac{12!}{(12-2)!}\). Then, we'll calculate the number of ways to distribute the remaining 10 books among 15 students, which is \(15P10 = \frac{15!}{(15-10)!}\).
04

Evaluating the permutation for part (b)

Substitute the values into the equation and compute the result for each permutation. Multiplying the result from the two permutations together, we find that the teacher can distribute the books in \(8789181600000\) ways such that the oldest student receives two books and no other student receives more than one book.

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Most popular questions from this chapter

The production of a machine part consists of four stages. There are six assembly lines available for the first stage, four assembly lines for the second stage, five for the third stage, and five for the last. Determine the number of different ways in which a machine part can be totally assembled in this production process.

The expansion \(\sum_{-3}^{7} \sum_{i=1}^{4} i j\) is an example of a double sum (or double summation). Here we find that \(\sum_{j=3}^{7} \sum_{i=1}^{4} i j=\sum_{i=3}^{7}\left(\sum_{i=1}^{4} i j\right)=\sum_{j=3}^{7}(j+2 j+3 j+4 j)=\sum_{j=3}^{7} 10 j\), after we expand the inner sum(mation) for the variable \(i\). Continuing, we then expand the outer sum(mation) for the variable \(j\) and find that \(\sum_{j=3}^{7} 10 j=10 \sum_{j=3}^{7} j=10(3+4+5+6+7)=\) 250. Hence \(\sum_{j=3}^{7} \sum_{i=1}^{4} i j=250\). Determine the value of each of the following double sums. a) \(\sum_{i=1}^{4} \sum_{j=3}^{7} i j\) b) \(\sum_{i=0}^{4} \sum_{j=1}^{4}(i+j+1)\) c) \(\sum_{j=1}^{4} \sum_{i=0}^{3} i\)

In how many ways can the letters in WONDERING be arranged with exactly two consecutive vowels?

A machine has nine different dials, each with five settings labeled \(0,1,2,3\), and 4 . a) In how many ways can all the dials on the machine be set? b) If the nine dials are arranged in a line at the top of the machine, how many of the machine settings have no two adjacent dials with the same setting? c) How many machine settings in part (b) use only 0,2 , and 4 as dial settings?

How many different paths in the \(x y\) plane are there from \((0,0)\) to \((7,7)\) if a path proceeds one step at a time by going either one space to the right \((R)\) or one space upward \((U)\) ? How many such paths are there from \((2,7)\) to \((9,14)\) ? Can any general statement be made that incorporates these two results?
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