find the number of distinguishable permuations in the word mississippi then the word hippopotamus
Consider the word Mississippi. It has 11 letters. There are 11! = 39,916,800 ways that the 11 letters can be rearranged, but that must be divided by the 4! = 24 indistinguishable arrangements of four s's, 24 indistinguishable arrangement of i's and 2 indistinguishable arrangements of p's.
11!/(4!*4!*2!) = 34,650
for hippopotamus i got 39,916,800 is that right?
To find the number of distinguishable permutations in a word, we need to count the number of ways we can rearrange the letters while considering any repeating letters.
For the word "mississippi":
1. Count the frequency of each letter:
- "m" occurs 1 time
- "i" occurs 4 times
- "s" occurs 4 times
- "p" occurs 2 times
2. Calculate the total number of permutations using the formula:
Total permutations = (total number of letters)! / ((frequency of first letter)! x (frequency of second letter)! x ... x (frequency of nth letter)!),
where n is the total number of distinct letters.
For "mississippi," we have:
- Total letters: 11
- Distinct letters: 4 (m, i, s, p)
- Frequencies: (1, 4, 4, 2)
Total permutations = (11)! / ((1)! x (4)! x (4)! x (2)!)
Now let's calculate this:
Total permutations = (11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / ((1 x 2 x 3 x 4) x (1 x 2 x 3 x 4) x (1 x 2 x 3 x 4) x (1 x 2))
Simplifying the expression:
Total permutations = 39,916,800 / (24 x 24 x 24 x 2)
Total permutations = 39,916,800 / 23,040
Total permutations ≈ 1,725
Therefore, there are approximately 1,725 distinguishable permutations in the word "mississippi."
Now let's move on to the word "hippopotamus":
1. Count the frequency of each letter:
- "h" occurs 1 time
- "i" occurs 1 time
- "p" occurs 3 times
- "o" occurs 3 times
- "t" occurs 1 time
- "a" occurs 1 time
- "m" occurs 1 time
- "u" occurs 1 time
- "s" occurs 1 time
2. Calculate the total number of permutations using the formula:
Total permutations = (total number of letters)! / ((frequency of first letter)! x (frequency of second letter)! x ... x (frequency of nth letter)!)
For "hippopotamus," we have:
- Total letters: 11
- Distinct letters: 9 (h, i, p, o, t, a, m, u, s)
- Frequencies: (1, 1, 3, 3, 1, 1, 1, 1, 1)
Total permutations = (11)! / ((1)! x (1)! x (3)! x (3)! x (1)! x (1)! x (1)! x (1)! x (1)!)
Now let's calculate this:
Total permutations = (11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / ((1) x (1) x (3 x 2 x 1) x (3 x 2 x 1) x (1) x (1) x (1) x (1) x (1))
Simplifying the expression:
Total permutations = 39,916,800 / (1 x 1 x 6 x 6 x 1 x 1 x 1 x 1)
Total permutations = 39,916,800 / 36
Total permutations ≈ 1,109,355
Therefore, there are approximately 1,109,355 distinguishable permutations in the word "hippopotamus."
To find the number of distinguishable permutations in a given word, we need to consider the frequency of each letter in the word.
1. For the word "mississippi":
- There are 11 letters in total.
- Frequency of 'm' = 1, 'i' = 4, 's' = 4, 'p' = 2.
- Using the formula for the number of distinguishable permutations, we can calculate it as:
N! / (F1! * F2! * ... * Fn!)
Where N is the total number of letters, and F1, F2..Fn represents the frequency of each letter.
So, for "mississippi":
Total permutations = 11! / (1! * 4! * 4! * 2!)
= 34,650
Therefore, there are 34,650 distinguishable permutations in the word "mississippi".
2. For the word "hippopotamus":
- There are 12 letters in total.
- Frequency of 'h' = 1, 'i' = 1, 'p' = 3, 'o' = 3, 't' = 1, 'a' = 1, 'm' = 1, 'u' = 1, 's' = 1.
- Using the same formula as above:
Total permutations = 12! / (1! * 1! * 3! * 3! * 1! * 1! * 1! * 1! * 1!)
= 199,584
Therefore, there are 199,584 distinguishable permutations in the word "hippopotamus".