Step-by-step explanation:
The letters of the word troposphere can be sorted to be:
ee h oo pp rr s t
with 7 distinct letters of which 4 are in pairs for a total of 11 letters.
The number of words that can be formed from n distinct letters
= n!/(1!1!1!....1!) [n times in the denominator]
The number of words that can be formed from n letters of which 2 are duplicated and the rest distinct is
= n!/(2!1!1!....1!) [n-1 times 1!]
Similarly, the number of words that can be formed from 11 letters, of which there are 4 pairs of duplicated letters is
N=11! / (2!2!2!2!1!1!1!)
= 39916800 / (2*2*2*2*1*1*1)
= 2494800