Praise be to Allah.
It is permissible to give a donation to a specific college or institute in return for the donor being let off having to take the entrance exam, and it is permissible for him to increase his donations so as to increase his chances of admission to a course of study.
That is because the college, in that case, has stipulated that, in order to be admitted, the student must either have a certain level of knowledge, or be willing to benefit the institution by giving money. This is permissible and there is nothing wrong with it.
Even though the money that is given is called a donation, in fact it is not a donation, because what is meant by a donation is that one gives money to someone for nothing in return, but this money that was given to the college is for something in return, which is admission to a course of study. So it is as if this student has purchased the right to study in this college by giving this money, so it is not a pure donation.
So if we assume that it is called a donation or a gift, it comes under the heading of a gift in return for something, which is permissible according to the correct scholarly opinion, like buying and selling.
See: al-Fawaakih ad-Dawaani (2/158)
It says in al-Mawsoo‘ah al-Fiqhiyyah al-Kuwaitiyyah (15/16):
What is meant by reward for a gift is financial recompense:
The basic principle concerning gifts is that there should not be any tangible recompense for them, because a gift is a donation, so it is not in return for something.
However it is permissible to give something in return for a gift, and this is called a reward for a gift. This refers to a gift for which recompense is received. The recompense for the gift may or may not be stipulated in the contract.
If it is stipulated in the contract, and it is known, then the contract is valid according to the Hanafis, Maalikis and Hanbalis, and according to the Shaafa‘is in the more well-known view, based on the meaning according to their view. The second view among the Shaafa‘is is that the contract is invalid because of the reference to compensation, because the word gift implies that it is a donation.
If the contract is valid then is a kind of transaction, or it is a transaction in general terms, and it comes under the rulings on transactions. In that case there is the right to go ahead with it or cancel it, or to return the item if it is faulty, and the right of first refusal (shuf’ah), and there is no right to return the item after the transaction has been completed. End quote.
It should be noted that this is on condition that the course of study in this college should be permissible, such as study of medicine, engineering, science or history and the like, and not study of something prohibited.
As for prohibited studies, such as study of singing, acting, music and the like, it is not permissible for a Muslim to join such courses or to give money to the college that teaches them.
And Allah knows best.