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How many different words can be formed using all the letters t

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How many different words can be formed using all the letters the word HONOLULU if no two alike letter are together.

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How many different words can be formed using all the letters the word HONOLULU if no two alike letter are together.

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Text solutionVerified

H = 1

;

N = 1

;

O, L, U = 2

Total permutations

= \frac{8 !}{2 ! 2 ! 2 !} = 7!

With both

O

together (7 items)

= \frac{7 !}{2 ! 2 !} = α

(let)

Similarly for both

L

together

= \frac{7 !}{2 ! 2 !}

also for

U = \frac{7 !}{2 ! 2 !}

Now with both

O

and

L

together

= \frac{6 !}{2 !} = β

With both

O

and

U

together

= \frac{6 !}{2 !}

with

U

and

L

together

= \frac{6 !}{2 !}

All three together

= 5! = y

Total

3 α - 3 β + y

Words with none together

= 7! - (3 α - 3 β + y)

= 7! - (3 (\frac{7 !}{2 ! 2 !} - \frac{6 !}{2 !}) + 5!)

7! - 5! (9 \times \frac{5}{2} + 1)

= 5! (42 - \frac{45}{2} - 1)

= \frac{120 ( 84 - 45 - 2 )}{2} = 2220

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Question Text	How many different words can be formed using all the letters the word HONOLULU if no two alike letter are together.
Answer Type	Text solution:1
Upvotes	161

How many different words can be formed using all the letters the word HONOLULU if no two alike letter are together.

Text solutionVerified

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