Which is grammatically correct? Or are they both OK?

A lot of people don't care about what kind of job they are doing.

A lot of people don't care about what kinds of jobs they are doing.
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And a third situation : Emotion: smile

A lot of people don`t care what kind of jobs there`re doing.

To differentiate, try to think at : That kind of job is illegal / I always try to avoid this kind of jobs / These kinds of jobs are exactly what people strive to avoid.
So even when the subject is the plural 'people', it doesn't have to be the plural 'what kinds of jobs'?
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It is not necessary. Here, the subject "people" requests plural only from the immediate verb which is "don`t care" . Anything logical can follow them.
A lot of people don't care about what kind of job they are doing.

A lot of people don't care about what kinds of jobs they are doing.

I would, however, keep the bolded number consistent. It is good style if nothing else.
OK. So both are fine but the second one is better. I see.

Thanks, MM!
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No, I just mean kind + job and kinds + jobs, not kind + jobs.

The choice of people + kind + job or people + kinds + jobs depends much on common sense rather than rule or style. If the writer/reader presumes that each person has only one kind of job, then perhaps people + kind + job is clearer.
Mister Micawber If the writer/reader presumes that each person has only one kind of job, then perhaps people + kind + job is clearer.
So is it what is called distributive singular?
Yes, that's the topic. Here is Quirk (quoted secondhand, I admit, from another site):

Quirk et al recognize that sometimes there is a number choice. They state this:

While the ....plural is the norm [in agreeing with a plural subject], the....singular may also be used to focus on individual instances. We therefore often have a number choice:

"¢ The children raised their hand / hands.
"¢ Some children have an understanding father / understanding fathers.
"¢ We all have a good appetite / good appetites....

The singular is sometimes obligatory or preferable with idioms and metaphors:

"¢ We are keeping an open mind [NOT keeping open minds]
"¢ They vented their spleen [NOT spleens] on him.
"¢ They can't put their finger [NOT fingers] on what's wrong.

The singular is sometimes used to avoid ambiguity:

"¢ Students were asked to name their favorite sport. [NOT sports if there is just one to be named by each student]
"¢ Children must be accompanied by a parent [NOT by parents if only one parent is required].
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