Assistant is idiot. Service counts from first date of joining and 3rd completes one day before joining date of that year.

Your 3 years are compete

 Study leave may be granted to a Government servant- (Rule 50 of CCS refers)

(i) who has satisfactorily completed period of probation and has rendered not less than five years’ regular continuous service including the period of probation under the Government;

(ii) who is not due to reach the age of superannuation from the Government service within three years from the date on which he is expected to return to duty after the expiry of the leave; and

(iii) who executes a Bond as laid down in Rule 53(4) undertaking to serve the Government for a period of three years after the expiry of the leave.

However since you have not completed 5 years of service, you may be eligible for EOL. 

In  Central Administrative Tribunal - Ernakulam

You are applicable for EOL it's before applying 3 years. 

After completing 3 years of continuous service, An employee will be eligible for extra ordinary leave. 

Extraordinary leave is granted to a Government servant when no other leave is admissible or when other leave is admissible, but the Government servant applies in writing for extraordinary leave.

2) ask the dealing assistant to furnish inspection of  rules wherein it is mentioned that to be eligible for EOL you had to complete 3 years on date of giving exams 

Dear Sir,

This is question of existing rule in relation to further/higher studies while in service and in some cases dealing assistants indulge in malpractice or misinterpret the rule. So, you are suggested to submit an humble application to your department/HR for seeking permission to join the said study and EOL for it. It is a natural law that no one will be stopped in any efforts for his career upgradation and even if some there is some administrative hurdles, the same may be relaxed. If the department/HR is not responding or submitting some contrary view, you are suggested to approach High Court under writ jurisdictions for such a direction to the department. 

Respected sir...

There is clearly mentioned that leave cannot be claimed as a matter of right and the leave sanctioning authority may refuse or revoke leave of any kind and Depending upon the nature and purpose for which the period of leave is to be availed of, EOL without pay and allowances will be granted only after the completion of a qualifying minimum service of 5 years at any institute ...

Thank you

Check the Rules-

7.1.12. Extra-Ordinary Leave (EOL)
i. Extraordinary Leave shall always be without leave salary and may be
granted when no other kind of leave is admissible, or when other leave
being admissible, the staff concerned has specifically applied in writing
for the grant of EOL.
ii. The period of EOL shall not count for increment except when such leave
is granted due to sickness on medical certificate or for prosecuting higher
studies, provided that in case of any doubt as to whether the EOL taken
was for prosecuting higher studies or not, the decision of the Board shall
be final.
iii. Except in the case of permanent staff, the duration of EOL on any one
occasion shall not exceed the following limits:
a. Three Months
b. Six months, where the staff has completed 3 years continuous
service on the date of expiry of the leave admissible to
him/her under the rules and his/her request for such leave is
supported by Medial Certificate.
c. Eighteen months, where the staff is suffering from TB,
Leprosy, Cancer or Mental illness and undergoing treatment
in a recognized clinic or under a specialist.
iv. EOL may also be granted to regularized periods of absence with out
leave retrospectively.
v. Depending upon the nature and purpose for which the period of leave is
to be availed of, EOL without pay and allowances will be granted only
after the completion of a qualifying minimum service of 5 years at this
Institute. It may be availed of for any one of the following purposes:
a. EOL for regular appointment.
b. EOL along with or without leave at credit for carrying
out research at higher levels in R & D organizations,
universities, etc.
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c. EOL with or without leave at credit for the purpose of
availing of research fellowships and other similar
activities.
d. EOL along with or without leave at credit for pursuing
higher studies leading to the award of a degree.
e. At any given time EOL (including leave at credit)
availed of will be for a minimum period of six months.
However, for availing of academic fellowship, the
Institute may permit EOL for a period of upto 90 days.
vi. EOL granted to staff members for short-term / long-term assignments in
India or abroad may be regulated on Foreign Service Terms (FST) for
the limited purpose of annual increments and terminal benefits. Pension
and leave salary contribution as per rules shall be paid either by the
employer or by the staff members concerned to NITT during the period
of Foreign Service. These contributions will entitle the staff member to
count his period of Foreign Service towards Pension, Annual Increment
and Earned Leave/HPL. However, such period of EOL will not count
towards continuity of service for Sabbatical Leave eligibility.
vii. For the staff members who do not wish to utilize Foreign Service terms,
extra-ordinary leave may be granted under the existing rules and they
will not be eligible for counting of service and will not earn leave for
that period and will also not be eligible for notional increment.
viii. The period of EOL availed for the purpose of prosecuting higher studies
leading to award of degrees or for the purpose of research fellowships
(not salaried jobs) will count for annual increments and qualifying
service for terminal benefits as well. EOL availed of for all other
purposes will not be counted for terminal benefits unless they are
regulated on Foreign Service terms.
ix. A staff member will be eligible to avail himself of a maximum of 5 years
of EOL without pay and allowance during the entire period of his service
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at this Institute (including periods of service elsewhere if these are taken
into account for retirement benefits).
x. For all purposes other than for prosecuting higher studies leading to the
award of a degree, a staff member will be eligible for EOL at the rate of
one year for every 5 years of qualified service put in at this Institute. If a
staff member has joined NITT after previous service at other NITs /
Central Universities, a maximum of 2 years can be added towards the
qualifying service for the purpose of working out the eligibility for long
leave other than Sabbatical Leave, provided such services at other NITs /
Central Universities were transferred to this Institute.
xi. On any single occasion a staff member can avail himself of EOL
(including leave at credit that may be attached) for a maximum of 2
years or the eligible period as per 1:5 norms, whichever is less.
xii. There must be a minimum interval of 3 years between two consecutive
periods of long leave.(i.e.) whose duration exceeds 6 months including
sabbatical leave.
xiii. The eligible period of EOL (excluding leave at credit) shall be calculated
as follows:
a. total period of service from the date of joining the Institute = n
years
b. total period of Special Leave and Sabbatical Leave already
availed of (excluding leave at credit = a years)
c.Total period of EOL availed of other than for higher studies
and on medical grounds and excluding leave at credit = b
years
d. EOL availed of for higher study = c years
e. Eligible period of EOL (excluding leave at credit)
is [n-(7a – 6b – c )]/5 or 5 years whichever is less.
The above period may be rounded off to a month.
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xiii. Whenever senior faculty/professors of the Institute go on deputation
to a higher position in an Institute of National importance, he/she can
be allowed for a maximum of 5 years and in all other cases they
should not be allowed for more than two years at a stretch. After
completing the first term in service ranging from 3 to 5 years
depending upon the nature of the initial contract, they should join
back the Institute. In case they get a second term of their contract
they should be willing to resign or retire from the services of the
Institute before taking up second term of the office.
xiv. The faculty requests on EOL for taking up assignments within
India/Abroad will be considered based on the following:
An internal committee will be formed by the Director to look into the
EOL requests of the faculty member. The committee will take in to
account that at any time, only one faculty members per department
could be spared on long leave (EOL), so that the teaching activity
would not be affected. Faculty appointed as Director / ViceChancellor to Government Institutions, would not be counted for the
said upper limit of one faculty per department. Long leave shall be
sanctioned only for appointments to Institutions of comparable (to
NITT) or superior standing. It is essential to ensure that the faculty
members add value to NITT, upon their return from such
appointments. The Director shall be empowered to take a final
decision in this matter (10th BoG dt. 23.12.2006).

- As per rule Study Leave is granted to all government employees who have more than 5 years of service.

- Further, the Extra Ordinary Leave (EOL) is granted to employees when no other leave is applicable and the employee specifically requests for the same in writing, if the employee has completed 3 years continuous service on the date of leave admissible 

- Since, you have joined 2nd January, 2017 , hence three years period completed on 3rd January ,2020, and hence you are eligible for EOL. 

- You should give a written complaint to your department head , to furnish and explain the rules for the counting of period for the same.  

The CCS rules arw silent as to the start of extraordinary leave. Now the ESIC rules may shed some light upon the start of eol. Kindly obtain the departmental rules.

A Government Employee is entitled to EOL on completion of three years from date of joining (DOJ)

The basic requirement is you need to make an application in writing, which needs to be considered by your Reporting Officer.

Your DOJ is 2.02.2017 and you have completed three years on 1.02.2020, as such, you are entitled to EOL.

If the Dealing Assistant creates any problem, you can insist him to share copy of rules based on which it can be taken forward, inspite of the above, if problem persists you may have to initiate legal proceedings.

You have completed 3 yrs of service so here you will be eligible to apply.

1. You ask him to provide the copy of rules under which this clause is mentioned that one have to complete three years before giving exam for post graduation.

2. If there is no such rule then you should contact next senior officer for grant of EOL.