# In how many different ways can the letters of the word 'LEADING' be arranged?

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In how many different ways can the letters of the word 'LEADING' be arranged in such a way that the vowels always come together?

posted May 19, 2014

Vowels-- E,A,I
Consonants-- L,D,N,G

Make a group of all vowels they are arranged in 3! ways.
Rest 4 consonants+1 group of vowels are arranged in 5! ways.

Total no. of ways.=3! *5!
=6*120
=720

Select the three vowels, and arrange them in 3! = 6 ways (EAI), (EIA), (IEA), etc.

Then, arrange the four consonants in 4! = 24 ways.

Insert the group of vowels in five ways: either to the left or the right of the consonants, or in any of the three spaces between two consonants. (For instance, if the vowel-group is arranged as AEI, and the consonant-group is arranged as LDNG, the groups may be arranged in five ways as (AEILDNG, LAEIDNG, LDAEING, LDNAEIG, LDNGAEI.)

3! × 4! × 5 = 720

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