correct answer is: a) 50,000 soldiers

Explanation

Let, there were ‘x’ soldiers in the army.
Now, \large\frac{3}{5} of soldiers in the army was killed.
\therefore Number of soldiers killed \left(x\times\large\frac{3}{5}\right)
\rightarrow \large\frac{3x}{5}
Now, \large\frac{1}{5} of soldiers in army was captured.
\therefore Number of soldiers captured \left(x\times\large\frac{1}{5}\right)
\rightarrow \large\frac{x}{5}
And, 10,000 soldiers fled away.
According to the question,
x=\large\left(\frac{3x}{5}+\frac{x}{5}\normalsize+10,000\right)
\rightarrow x=\large\left(\frac{3x+x+50,000}{5}\right)
\rightarrow 5x=\left(3x+x+50,000\right)
\rightarrow 5x=4x+50,000
\rightarrow 5x-4x=50,000
\rightarrow x=50,000
\therefore There were 50,000 soldiers in the army.
Ans: There were 50,000 soldiers in the army.

An additional variation of this particular math query can be considered:

1. Calculate the total number of soldiers in the army given that 3/5 were killed, 1/5 were captured, and the remaining 10,000 fled away.
2. Find out the original size of the army when 3/5 were killed, 1/5 were captured, and the rest fled away, leaving 10,000.
3. Calculate the initial size of the army when portions were lost in battle, and 10,000 fled after 3/5 were killed and 1/5 were captured.
4. Determine the total army strength when fractions were lost in battle, with 10,000 fleeing after 3/5 were killed and 1/5 were captured.