# Permutations and combinations

## A group contains n men and n women. How many ways are there to arrange these people in a row if the men and women alternate?

Relevant definitions to solve this exercise: “THE PRODUCT RULE Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there […]

## There are six different candidates for governor of a state. In how many different orders can the names of the candidates be printed on a ballot?

Let’s refresh the definitions. “A permutation of a set of distinct objects is an ordered arrangement of these objects. We also are interested in ordered arrangements of some of the elements of a set. An ordered arrangement of r elements of a set is called an r-permutation. ” “An r-combination of elements of a set

## In how many different orders can five runners finish a race if no ties are allowed?

First, let’s examine the definitions of permutations and combinations. “A permutation of a set of distinct objects is an ordered arrangement of these objects. We also are interested in ordered arrangements of some of the elements of a set. An ordered arrangement of r elements of a set is called an r-permutation. ” “An r-combination

## Find the value of each of these quantities a) C(5,1) b) C(5,3) c) C(8,4) d) C(8,8) e) C(8,0) f) C(12,6)

This is a simple exercise. To solve it, we just need to apply the formula to calculate the number of r-combinations. “The number of r-combinations of a set with n elements, where n is a nonnegative integer and r is an integer with 0 ≤ r ≤ n, equals C(n,r) = n! / r!(n−r)!” Discrete

## Find the number of 5-permutations of a set with nine elements

This is another simple exercise. To solve it, we just need to apply the formula to calculate the number of 3-permutations. “If n and r are integers with 0 ≤ r ≤ n, then P(n,r) = n! / (n−r)! ”. Discrete mathematics and its applications by Rosen. Answer: We just need to calculate the number

## Find the value of each of these quantities

This type of exercise is easy. You are only required to apply a formula. You just need to know the formula. “If n and r are integers with 0 ≤ r ≤ n, then P(n,r) = n! / (n−r)! ”. Discrete mathematics and its applications by Rosen. a) P(6,3) P(6,3)=6!/(6-3)! =6!/3! =6x5x4x3!/3! =120 b) P(6,5)

## How many permutations of {a, b, c, d, e, f, g} end with a?

“A permutation of a set of distinct objects is an ordered arrangement of these objects.” We can follow a simple approach to solve this exercise. If we calculate how many permutations are there in the set {b, c, d, e, f, g} (the original set without ‘a’), we will know how many permutations of the

## A professor writes 40 discrete mathematics true/false questions. Of the statements in these questions, 17 are true. If the questions can be positioned in any order, how many different answer keys are possible?

A powerful tool for solving problems is to transform a given problem into another one that is easier to solve, or that we already know how to solve. Then, from the solution to that new problem, we can give the solution to the former one. This is such a case. We know that we have

## Let S = {1,2,3,4,5}. List all the 3-permutations of S. List all the 3-combinations of S.

“A permutation of a set of distinct objects is an ordered arrangement of these objects.” Theorem: “If n is a positive integer and r is an integer with 1 ≤ r ≤ n, then there are P (n, r ) = n(n − 1)(n − 2) · · · (n − r + 1) r-permutations

## How many different permutations are there of the set  {a,b,c,d,e,f,g}?

This type of exercise is quite easy, and probably you won’t find it commonly on a test, exam, or in real life. However, there are some places you might find it. So, it is worth looking into it. As usual, let’s start with relevant definitions and theorems. “A permutation of a set of distinct objects