れいぞうこ の なか に ケーキ は 3こ ありました.
せんしゅう 、かんじ を 20こ おぼえ ました.
In the above examples, why does こ come after the number? Is it a counter? If yes, at what instances can it be used? If not, please advise how and when it should be used.
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The こ is indeed a counter, usually written 個.
It is a general counter for counting objects, much like つ. Your two example sentences provide proof of the generality of 個, since cakes and kanji characters have few common features.
つ is often taught as the "default" counter ("default" meaning that it can be used in practically any situation to make yourself understood). But つ is only used up to 10 objects. For 11 or more objects, 個 could be regarded as the default counter.