How many arrangements can be made from the letters of the word "mountains" if all the vowels must form a sting?

Question

I thought we take $4$ vowels and find number of arrangements $4!$ and multiply it with arrangements that can be made with consonants that is $5!/2!$. However my approach seems to be wrong.

for every $5!/2$ arrangements ,for example .M_N_T_N_S. thereis 6 gaps to put $4!$ arrangements of vowels — serg_1, Jun 15 '17 at 07:23

score 3 · Accepted Answer · answered Jun 15 '17 at 07:19

It's almost right. That counts the number of ways to (say) start with all the vowels, and then have all the consonants.

But for your question, once you've chosen the order of the vowels and the order of the consonants, there are $6$ positions you could put the vowels in (before all the consonants, after the first consonant, and so on to after all the consonants). So you have to multiply by $6$.

score 3 · Answer 2 · answered Jun 15 '17 at 07:20

Hint. We have $9-4+1=6$ positions for the initial vowel. The number arrangements of the $4$ vowels is $4!$ (they are distinct). The number arrangements of the $5$ consonants is $5!/2!$ (due to the double "n"). So what is the total number of arrangements?

How many arrangements can be made from the letters of the word "mountains" if all the vowels must form a sting?

2 Answers2