How many distinct four-letter words beginning with A can be formed from letters with two similar letters and two different letters?

Question

Four-letter words are made from letters A, A, D, E, E, M, S, Y such that two letters are similar & another two are different & each word begins with letter 'A'. The total number of such words will be

a) $\ 60 \quad $ b) $\ 80\quad $ c) $\ 100\quad $ d) $\ 120$

My try:

Distinct letters: A, D, E, M, S, Y

Since each word begins with 'A' hence word structure will be $\ \boxed{A} \boxed{X}\boxed{X}\boxed{X}$

If we take another 'A' then rest three places can be filled by total

$=3\times 5\times 4$

$=60$

If we take two 'E' then rest three places can be filled by total

$=3\times3 \times 4$

$=36$

Total number of required words of four letters

$=60+36$

$=96$

but there is no option for $96$. My answer is wrong. My teacher says that option (d) 120 is correct answer. But I don't know how. Somebody please help me solve this problem. Thanks

Answer 1

If we take two 'E' then rest three places can be filled by total =3×3×4 =36

(take two E)(another one Q other than A)(choose a position for Q) so

$$(1)\cdot({|\{D,M,S,Y\}| \choose 1})\cdot({3\choose1})=4\cdot3=12.$$

Why you get 36?