How many ways can the 5 letters 'MANNA' arranged in a row?

a)
There are 2 vowels (A, A) and 3 consonants (M, N, N).
The row are _ _ _ _ _.

Firstly, select 2 letters from the 3 consonants and put the 2 letters into the first and last positions.
No. of ways = ₃P₂/2! = 3!/2! = 3
(It is divided by 2! because there are 2 N's.)

Second, put the 3 letters left into the second, third and fourth positions.
No. of ways = ₃P₃/2 = 3!/2! = 3
(It is divided by 2! because there are 2 A's.)

Total no. of ways = 3 × 3 = 9

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b)
Firstly, put the 2 A's into the second and fourth positions, i.e. _ A _ A _
No. of ways = 1

Secondly, put the three consonants into the first, third and last positions.
No. of ways = ₃P₃/2 = 3!/2! = 3
(It is divided by 2! because there are 2 N's.)

Total no. of ways = 1 × 3 = 3

a) There are 3 ways to make the first and last letter consonants: M***N, N***N and N***M

For each of these, there are 3 letters in the middle so there are 3! = 6 ways to arrange 3 middle letters. However since A is duplicated it is double-counted, so in fact there 6/2 = 3 ways to arrange the middle letters.

Total = 3*3 = 9

Here they are:
MANAN, MAANN, MNAAN
NANAM, NAANM, NNAAM
NAMAN, NAAMN, NMAAN

b) Since there are 3 consonants and only 2 vowels, we must always start with a constant.  The simplest method is just to list all possibilities:
MANAN,NAMAN,NANAM
There are 3 arrangements.