In this video: https://youtu.be/HxctJzSADPI?t=2m18s there appears the sentence "Manchen Leuten macht Mathe Spaß." Since "Leute" is already plural, I don't understand the usage of "Leuten" here.
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You are right, Leute is a plurale tantum, i.e. a word that only exists in the plural. But that does not mean that it can't be declined.
This is the declension of »Leute«:
- nominative: die Leute (Die Leute schlafen.) (The people sleep. or: The people are sleeping.)
- genitive: der Leute (Die Kleider der Leute sind schön.) (The people's clothes are beautiful.)
- dative: den Leuten (Die Kleider gehören den Leuten.) (The clothes belong to the people.)
- accusative: die Leute (Ich sehe die Leute.) (I see the people.)
As you can see, in dative case there is an extra n at the end of Leute: den Leuten.
Now let's analyse your example sentence:
manchen Leuten = dative object
macht = predicate
Mathe = subject (in nominative case)
Spaß = accusative object
Maybe you are used to have the subject on position 1 of the sentence, like in this sentence:
Mathe macht manchen Leuten Spaß.
This is correct too, but German has a much more flexible word order, so, as long as you leave the predicate on position 2, you can do almost everything. (Well, not really everything, but much more than in english.) In German you don't identify the logical parts of a sentence by its position within the sentence, but by its grammatical case.
Der Bauer verfolgt den Stier. = The farmer chases the bull.
Den Bauer verfolgt der Stier. = The bull chases the farmer.
Den Stier verfolgt der Bauer. = The farmer chases the bull.
Der Stier verfolgt den Bauer. = The bull chases the farmer.
The subject in your sentence is the thing that makes fun, and this of course is mathematics. So »Mathe« (short form of »Mathematik«) is the subject and therefore has to be in nominate case.
»Spaß« (fun) is the thing that is made by mathematics. An object that is made by something else has to be used in accusative case (the verb »machen« demands accusative case), and this object is fun (Spaß).
People (die Leute) is the object to whom something is made. They receive the fun. And when ever an object receives something you use dative case (the name dative comes from latin »do« which means »to give«, so dative is the giving-to-case). So in German you have to say »den Leuten«.