There are a few uses of the number "10" in this passage.
We are in Volume III, Part One, Chapter XXII. Just before the fall of Moscow. The nobility is gathered together and they are asked to donate ten men per thousand to serve as soldiers. This is prior to the freeing of the serfs which took place in 1861 so presumably these "donations" are serfs. The sovereign gives a short speech of thanks to the nobility, then speaks with the merchants for about ten minutes. Pierre is present and is very moved. Pierre is so sentimental and easily influenced by others, so when he learns that one man has given a regiment, he gives a thousand men and their maintenance.
Natasha's younger brother has been wanting to enlist, and after witnessing the sovereign's speech, he not only agrees but helps Petya sign up. -- Lola
from page 101-102, Volume 2 of original text
Pevear/Volokhonsky translation page 680-682