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Chapter 11: Problem 70

Explain how to distinguish between permutation and combination problems.

Short Answer

Expert verified
The key distinction between permutation and combination problems is that the order of selected items matters in permutation, while it doesn't in combination problems.

Step by step solution

01

Defining Permutation and Combination

A permutation is an arrangement of objects in a specific order. For instance, how many ways can 3 students from a class of 10 be chosen to give a presentation? Here, order is important, and this is a permutation problem. On the other hand, a combination is a selection of items where ordering is not important. For example, considering a lottery draw of 6 numbers from 49. In this scenario, the order of the numbers doesn't matter; it's purely a selection issue, and hence this is a combination problem.
02

Use of Permutations and Combinations in Real-life scenarios

Now, apply these definitions to some real-life scenarios. For instance, arranging books on a shelf would require permutations because the order in which you place the books matters. Meanwhile, when pulling socks from a drawer, combinations are typically used because the order in which you pull the socks does not matter. For permutations, the formula is \(P(n, r) = n! / (n-r)!\). For combinations it鈥檚 \(C(n, r) = n! / {r!(n-r)!}\).
03

The Importance of Context in Problems

It's crucial to understand the context of the problem. If the sequence or arrangement of selected objects is important, then it's a permutation problem. If the order isn't important, then it's a combination problem. Remember the simple rule 鈥� If order matters, it's a permutation; if not, it's a combination.

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Most popular questions from this chapter

In the original plan for area codes in 1945 , the first digit could be any number from 2 through 9 , the second digit was either 0 or 1 , and the third digit could be any number except 0 . With this plan, how many different area codes are possible?

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. An apartment complex offers apartments with four different options, designated by A through D. There are an equal number of apartments with each combination of options. $$ \begin{array}{|l|l|l|l|} \hline \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \text { one bedroom } & \text { one } & \text { first } & \text { lake view } \\ \text { two bedrooms } & \text { bathroom } & \text { floor } & \text { golf course } \\ \text { three } & \text { two } & \text { second } & \text { view } \\ \text { bedrooms } & \text { bathrooms } & \text { floor } & \text { no special } \\ & & & \text { view } \\ \hline \end{array} $$ If there is only one apartment left, what is the probability that it is precisely what a person is looking for, namely two bedrooms, two bathrooms, first floor, and a lake or golf course view?

A stock can go up, go down, or stay unchanged. How many possibilities are there if you own seven stocks?

We return to our box of chocolates. There are 30 chocolates in the box, all identically shaped. Five are filled with coconut, 10 with caramel, and 15 are solid chocolate. You randomly select one piece, eat it, and then select a second piece. Find the probability of selecting two caramel-filled chocolates in a row.

Consider a political discussion group consisting of 5 Democrats, 6 Republicans, and 4 Independents. Suppose that two group members are randomly selected, in succession, to attend a political convention. Find the probability of selecting two Republicans.
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