Explanation:
so, we have 10 letters.
if the would be all different, then we would have
10!
different arrangements : 10 options for the first position, then 9 remaining options for the second position, then 8 remaining options for the third position, ...
but - we have 2 As, 2 Es, 2 Ms, 2 Ns.
that means that e.g. the original arrangement MANAGEMENT
looks the same, if we simply have the As trading places. or the Es, or the Ms, or the Ns.
formally, they are different arrangements, but we cannot distinguish them. for us observers they are all the same arrangement.
for each arrangement we have 2 identical looking ones based on the 2! possibilities to arrange the 2 letters A. the same for Es, Ms and Ns.
so, we need to cut the original options in half (divide by 2!) to compensate for the 2 As trading places. and that again in half for the Es. and that again in half for the Ms. and that again in half for the Ns.
so, we have
10! / 2 / 2 / 2 / 2 =
= 10! / 2⁴ = 10! / 16 = 226,800 different arrangements.
FYI - as indicated, formally we would have to divide by 2! every time, as we need to divide by the number of possibilities to arrange the identical letters (in the same way).
as 2 = 2! this does not make a difference.
but if we had a letter appearing e.g. 3 times in the original word, we would have to divide by 3! = 6 to eliminate all the duplicates based on this letter, because the 3 identical letters can be arranged in 3! = 6 different ways (and we could not tell the difference).