Question

How many arrangements of MASON are there such that S occurs before N and the vowels are in alphabetical order?

Answer 1

A should always come before O and S will always occur before N.

That means all the arrangements that begin with O or N will not be considered.

Hence, the arrangements should begin with either M, A, or S.

i) First letter M

The remaining 4 letters can't begin with either O or N. So the second letter should be either A or S.

If the second letter is A, the remaining 3 letters can be permuted in 6 ways. Out of those 6, only 3 will be acceptable. They are

MASON, MASNO, MAOSN

If the second letter is S, again only 3 cases will be acceptable.

MSAON, MSANO, MSNAO

In total, 6 arrangements are possible.

ii) First letter A

The remaining 4 letters can't begin with N. The second letter will be either S O or M.

If the second letter is S, all 6 permutations of the remaining letter are possible. No of ways = 6.

If the second letter is O, 3 arrangements are possible. They are

AOSNM, AOSMN, AOMSN

If the second letter is M, again 3 arrangements are possible. They are

AMSNO, AMSON, AMOSN

Total number of ways = 6 + 3 + 3 = 12

iii) First letter S

Similar to A, if the first letter is S, a total of 12 cases are possible.

Overall total arrangements possible is = 6 + 12 + 12 = 30

Required answer = 30

Thank You!! Please Upvote!!!