Number of Possible Arrangements (Permutations?)

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In summary, for a multiple-choice test with 5 questions and 4 answer choices for each question, there are 1024 ways for a student to answer all the questions (using the multiplication rule). And for part (b), if the student wants to get all the answers wrong, there are 243 ways to do so by choosing 3 out of 4 choices for each question.
  • #1
Saladsamurai
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Homework Statement


If a multiple-choice test consists of 5 questions each with 4 possible answers of which only 1 is correct,

(a) In how many different ways can a student check off one answer to each question?

(b) In how many different ways can a student check off one answer to each question and get all the answers wrong?


I feel like if I could get a hint on (a) I could do (b) as well, but I am a little stuck trying to figure out which rule to apply. These are permutations correct? (the arrangement of answers.)

Or should I just apply the "multiplication rule" somehow?
Can I get just a hint here?
 
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  • #2
a) How many ways are there to answer ONE question? i.e. How many ways are there to PICK/CHOOSE one of the four answers? How many ways are there to do this for 5 questions?
 
  • #3
The Chaz said:
a) How many ways are there to answer ONE question? i.e. How many ways are there to PICK/CHOOSE one of the four answers? How many ways are there to do this for 5 questions?

Let's denote each answer choice for each question as a, b, c, d.

So for each question there are 4 different ways to answer. Oh. So it is 4^5 = 1024. So it is the multiplication rule.
 
  • #4
So I suppose for part (b) since I have (4-1) choices for each, the answer would be (4-1)^5 = 243.
 
  • #5
Sounds good to me!
 
  • #6
Thanks homes!
 

What is the formula for calculating the number of possible arrangements (permutations)?

The formula for calculating the number of possible arrangements (permutations) is n! (n factorial), where n is the total number of items or objects.

How is this formula derived?

The formula for calculating the number of possible arrangements (permutations) is derived from the fundamental counting principle. This principle states that if there are n ways to do one thing and m ways to do another, then there are n x m ways to do both in sequence. Therefore, for a set of n items, there are n ways to choose the first item, (n-1) ways to choose the second item, (n-2) ways to choose the third item, and so on, resulting in n x (n-1) x (n-2) x ... x 3 x 2 x 1 = n! possible arrangements.

How is this formula applied in real-life situations?

The formula for calculating the number of possible arrangements (permutations) is used in various fields, such as mathematics, computer science, and statistics. It can be applied in situations where the order or sequence of items is important, such as in the arrangement of letters in a word, in shuffling a deck of cards, or in creating unique combinations of numbers for passwords.

What is the difference between permutations and combinations?

Permutations and combinations both involve counting the number of possible arrangements, but the main difference is that permutations take into account the order or sequence of items, while combinations do not. In other words, permutations are concerned with arrangements, while combinations are concerned with selections.

Are there any limitations to the formula for calculating the number of possible arrangements (permutations)?

Yes, there are limitations to this formula. It can only be used for distinct or unique items, and it assumes that all items are used in the arrangement. Additionally, if there are identical items in the set, the formula may overcount the number of possible arrangements. In such cases, a different formula, such as the multinomial coefficient, may be used.

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