How many ways we distribute 10 distinct objects among 4 persons such that only one person gets exactly 4 objects? 
How many ways we distribute 10 distinct objects among 4 persons such that only one person gets exactly 4 objects?

The correct answer given is ${10 \choose 4} \cdot 3^6$. Here they are choosing $4$ objects out of the $10$ and giving them to one person and the remaining $6$ objects have $3$ choices each.
Shouldn't the answer be ${10\choose 4}{4 \choose 1} \cdot 3^6$ as for the group made of $4$ objects we have $4$ choices too. So choose $4$ from $10$ objects and choose $1$ from $4$ persons and then give that $4$ objects to that person. Why is this approach wrong?
 A: Both your answer and the stated answer allow more than one person to receive exactly four objects, so both are wrong.
Given that the wording of the question does not specify that a particular person is to be the one to receive four objects, let's go with your interpretation of the question.
There are four ways to choose the person who is to receive exactly four objects, $\binom{10}{4}$ ways to select which four objects that person is to receive, and $3^6$ ways to distribute the remaining six objects to the remaining three people, which gives us a preliminary count of
$$\binom{4}{1}\binom{10}{4}3^6$$
However, this allows for the possibility that two people each receive exactly four objects.
We have counted each case in which two people receive exactly four objects twice, once for each way we could have designated one of those two people as the person who receives exactly four objects.  We do not want to count those cases at all, so we must subtract twice the number of those cases from the total.
There are $\binom{4}{2}$ ways to select two people to each receive exactly four objects, $\binom{10}{4}$ ways to select which four objects will be received by the elder of those two people, $\binom{6}{4}$ ways to select which four of the remaining six objects will be received by the younger of those two people, and $2^2$ ways to distribute the remaining two objects to the remaining two people.  Hence, there are
$$\binom{4}{2}\binom{10}{4}\binom{6}{4}2^2$$
distributions in which two people each receive exactly four objects.
Hence, the number of ways $10$ distinct objects can be distributed to four people so that exactly one person receives exactly four objects is
$$\binom{4}{1}\binom{10}{4}3^6 - 2\binom{4}{2}\binom{10}{4}\binom{6}{4}2^2$$
