Question

How many six letter permutations can be formed from the letters of the word hubbub

Answer 1

60

==============

The word "hubbub" consists of one "h," two "u"s, and three "b"s.

A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement.

Since we need to form six-letter permutations with all the letters from "hubbub", we have to calculate the permutations while considering the repeated letters.

The formula to find permutations of a set of objects when there are repeating items is:

• 
  `\[ P = \cfrac{N!}{n_1! \cdot n_2! \cdot ... \cdot n_k!} \]`
  

Where:
- N is the total number of objects
-
`n_1, n_2, ..., n_k` are the frequencies of each distinct object.

For "hubbub":
- We have a total of N = 6 letters
- "h" occurs once, so
`\( n_h = 1 \)`
- "u" occurs twice, so
`\( n_u = 2 \)`
- "b" occurs thrice, so
`\( n_b = 3 \)`

Now apply the formula:

• 
  `\[ P = \cfrac{6!}{1! * 2! * 3!} \]`

We calculate the factorial of each:

-
`\( 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720 \)`
-
`\( 1! = 1 \)`
-
`\( 2! = 2 * 1 = 2 \)`
-
`\( 3! = 3 * 2 * 1 = 6 \)`

Now plug these into the formula to get:

• 
  `\[ P = \cfrac{720}{1 * 2 * 6} \]`
• 
  `\[ P = \cfrac{720}{12} \]`
• 
  `\[ P = 60 \]`

So there are 60 different six-letter permutations.