In terms of expressing ten, what's the difference between десять, десятка and десяток? I understand the first is a numeral and the other two are nouns, but I'm curious to know in what contexts each is used.
Also, in the sentence Он сыграл в кино десятки ролей, is десятки a declined form of десятка or десяток?
Десятка - Typically, an informal reference to something numbered 10 (like a bus following route 10) or about a 10 ruble banknote (in the past, when it mattered more). Now, it can sometimes informally mean 10,000 rubles; or informally refer to a 10-year prison sentence (ему дали "десятку").
Десяток - 10 pieces (similar to a dozen eggs etc.), 10 years of age in expressions like "ему уже седьмой десяток" (he's already in his 60's). It can also be used to approximate time (прошёл/пройдёт десяток лет - один или более десятков). Mathematically, десятки (plural)refers to the tens slot in numbers.
P.S. In "Он сыграл в кино десятки ролей", десятки is the accusative plural form of десяток.
Also, in the sentence Он сыграл в кино десятки ролей, is десятки a declension of десятка or десяток?
this is a declined десяток
десятки is never meant in a literal sense of quantity but just used to indicate a great amount within certain limits — in this case under 100.
In this usage, as Alex_ander noted, it's used idiomatically to refer to blocks of ten, Cf. dozens.
Likewise cотни, тысячи, десятки тысяч and so on.
At the same time, десяток and the nouns just listed can indicate a precise number if qualified by a numeral
Он сыграл в кино два десятка ролей
or express uncertainty about the accuracy of the number (quantity) if followed by one (there's a more general syntactic rule at work here)
Он сыграл в кино десятка два ролей
So, in an attempt to summarize this, it might be said that десятка (and any other numerical cognate noun ending in -ка such as тройка, семёрка, двадцатка, пятидесятка and so on) can be applied to something solid and indivisible, whereas десяток - to a set of separate items of one kind.
The difference can be seen in the following sentence:
Я купил десяток (штук) десяток = Я купил десять (штук) десяток
where the 2nd десятка means an item numbered 10, for example a spool of sewing thread size 10, and so the entire sentence means I bought ten items of the (number) 10.
ДЕСЯТКА (coll.) fem. noun:
The name of the number "ten"
Colloquially, the name of a means of transport (a bus, a tram) which has the number 10
Какой автобус тут ходит? — Десятка.
Bull's-eye (sports) — the black circle in the center of a target, as in the idiom:
попасть в десятку — to hit the nail right on the head.
A ten-ruble coin
A card, e.g. the ten of spades — десятка пик, the ten of hearts — десятка червей.
ДЕСЯТОК masc. noun:
Ten things (Cf. dozen)
Десяток пирожных.
Decade (in age)
Ему пошёл шестой десяток — He's in his fifties.
A group of ten people
Десяток ребят.
In mathematics, the tens — тысячи, сотни, десятки, единицы.
A score of десятка два людей
Idioms
Не робкого десятка — He's brave.
Добрый десяток лет — Over a decade; a good ten years.
