How many arrangements are possible using the letters in the word FUZZY if each letter "Z" is distinctly different than the other? How many arrangements are possible if the letter "Z" is interchangeabe with the other? Explain your reasoning.
(a)
"Z" is distinctly different than the other: I think that means you have to count both of them, as if they're different letters.
There are five letters in total. Therefore,
5! = 120 ways
(b)
"Z" is repeated twice in the word FUZZY. Thus,
5! / 2! = 60 ways
hope this helps~ `u`
You use a factorial (!) with the number of articles given. A factorial is a product of an integer and all integers below it.
5 Factorial
5! = ___ combinations
To determine the number of arrangements possible using the letters in the word "FUZZY," we need to consider two scenarios:
Scenario 1: The letter "Z" is distinctly different from others.
Scenario 2: The letter "Z" is interchangeable with the other letters.
Scenario 1: Distinct "Z"
In this scenario, we treat each "Z" as a distinct letter, so there are two different "Z"s. The word "FUZZY" has a total of 5 letters.
To calculate the number of arrangements, we use the formula for permutations:
nPr = n! / (n - r)!
where n is the total number of letters and r is the number of letters we are arranging.
In this case, n = 5 and r = 5 (as we are arranging all 5 letters).
Substituting the values into the formula:
5P5 = 5! / (5 - 5)!
= 5! / 0!
= 5! / 1
= 5 x 4 x 3 x 2 x 1
= 120
Therefore, there are 120 possible arrangements when each "Z" is distinctly different from the other letters.
Scenario 2: Interchangeable "Z"
In this scenario, we treat both "Z"s as the same letter, and they can be interchanged without creating a different arrangement.
To calculate the number of arrangements in this case, we again use the formula for permutations.
However, since we now have 4 distinct letters ("F," "U," "Z," "Y"), the total number of letters for arranging (n) is reduced to 4.
Using the formula:
4P4 = 4! / (4 - 4)!
= 4! / 0!
= 4! / 1
= 4 x 3 x 2 x 1
= 24
Therefore, when the two "Z"s are interchangeable, there are 24 possible arrangements.
In summary:
- When each "Z" is distinctly different, there are 120 possible arrangements.
- When the two "Z"s are interchangeable, there are 24 possible arrangements.
To find the number of arrangements, we need to consider the number of distinct letters in the word and the repetitions.
1. If each letter "Z" is distinctly different from the other:
- The word "FUZZY" has 5 letters, with 2 "Z" and 1 each of "F", "U", and "Y".
- We can calculate the number of arrangements using the formula for permutations with repetitions allowed: n!/ (r1! * r2! * ... * rk!), where n is the total number of letters and r1, r2, ..., rk are the number of repetitions for each distinct letter.
- In this case, n = 5, r1 = 2, r2 = 1, r3 = 1, and r4 = 1.
- The number of arrangements will be 5! / (2! * 1! * 1! * 1!) = 5 * 4 * 3 = 60.
2. If the letter "Z" is interchangeable with the other:
- The word "FUZZY" still has 5 letters, but now we have 3 repetitions of the letter "Z" (as Z1, Z2, and Z3) along with 1 each of "F", "U", and "Y".
- Using the same formula as before, n = 5, r1 = 3, r2 = 1, r3 = 1, and r4 = 1.
- The number of arrangements will be 5! / (3! * 1! * 1! * 1!) = 5 * 4 = 20.
The key difference between the two cases is whether we treat the "Z"s as distinct or interchangeable. When they are distinct, each arrangement is unique because the "Z" at each position is different. However, when they are interchangeable, some arrangements will be identical due to the similarity of the "Z"s.
Oh, FUZZY arrangements? Sounds like a fuzzy math problem! Let's dive in.
If each letter "Z" is distinctly different from the other, the word FUZZY consists of 5 letters: F, U, Z₁, Z₂, and Y. To find the number of arrangements, we use the formula for permutations:
Number of arrangements = n!
Where n is the number of items to arrange.
In this case, n = 5 because there are 5 distinct letters. Therefore,
Number of arrangements = 5!
Calculating that, we get:
Number of arrangements = 5 × 4 × 3 × 2 × 1 = 120
So, there are 120 possible arrangements of the letters in FUZZY when each "Z" is distinctly different.
Now, let's talk about the interchangeable Z's. If the letter "Z" is interchangeable with the other letters, it means that there's only one "Z" in the word FUZZY. So now, the word consists of 4 distinct letters: F, U, Z, and Y.
Using the same formula, we get:
Number of arrangements = 4!
Calculating that, we get:
Number of arrangements = 4 × 3 × 2 × 1 = 24
Therefore, if the letter "Z" is interchangeable with the other letters, there are 24 possible arrangements.
So remember, whether it's distinctly different or interchangeable, FUZZY always keeps us on our toes!