I have friends pursuing courses you mentioned above at either institute. Based on the discussion I have with them, here is what should be best for you,

1. The institutes can be considered quite equivalent to one another, but BITS Goa has an upper hand. 
2. M.Sc at BITS Goa can be specialized following the first year, during which you will be allocated a B.E course depending upon your CGPA. 
3. With a 6 CGPA, you can get Mechanical Engineering, which is quite an easy job. In contrast, it is better than ICE.
4. However, if your scores are high enough then you might get a seat in CSE. 
5. By the end of 5 years, you will be receiving a B.E and an M.Sc degree, which is nonetheless a smarter choice. 

However, the fees at BITS Goa are quite a lot. 

The answer to the question may be crisp and short but a government college is better than a private college any day in India for higher education.

It is a comparison between two brands, BITS and NIT. The bigger and better brand is NIT when compared to BITS.

Apart from the great brand value of government colleges, studying and being graduated from a government-affiliated college has various advantages over other top private colleges. However, the option you prefer and want to pursue is completely your choice. Consider the parameters and opt for the best. 

There are different viewpoints for the same question. It primarily depends upon what is best for you, and whether or not the course you are willing to take interests you. 

One of my friends encountered a similar problem likewise, and addressing the same, here are some points you might want to consider before diving into any conclusion,

1. A dual degree in Mathematics, Physics, and Chemistry is claimed as a part of Science, while others are considered in the case of Engineering. You will learn the theories of science instead of the application of technology.
2. In Mathematics, you are supposed to learn the series of higher-order and their respective application. While, in Mechanical Engineering, you will learn more deeply about Engines, Boilers, Automobiles, Refrigerator Systems, and much more.
3. A single degree, like Mechanical Engineering, CSE, EE, Civil Engineering, or Chemical Engineering, can be more advantageous than dual degrees. 
4. If you opt for a dual degree in Mathematics, you have a higher probability of becoming a mathematician or a professor. But, in case you pursue Mechanical Engineering from NIT Surat, then you bear higher chances of securing a placement offer at top-tier MNCs.
5. Your choice should always remain ahead of most others. The one piece of advice is that in NIT Surat, you supposedly cannot change your degree from M.Sc to B.Tech following the 1st year and vice versa. 

You should research the same for BITS Goa and find whether or not the rule is similarly applicable in the institute. This is the basic difference you can find between both institutes. 

First of all, I want to clarify that there is no such branch called CSC in either BITS Pilani or NIT Calicut. Both the colleges have Computer Science & Engineering (CSE) degrees 

• BITS Pilani is a top private engineering university. However the same cannot be said about its Goa campus. 
• NIT is a brand that is recognized instantly by almost everyone in any part of the country and this gives NIT Calicut a distinct advantage. 
• As it is a government-run college the learning environment is also better due to the better quality of students. 
• Companies also prefer NITs over private engineering colleges. You will get better job opportunities at NIT Calicut.

Hence it is suggested that CSE at NIT Calicut is a much better choice than CSE at BITS Pilani Goa campus.

Presidency University, Kolkata is a better to place to study economics than BITS, Goa, or Hyderabad. Presidency University, Kolkata is well known for its economics department. It also has an excellent subject record.

In the national ranking too Presidency University comes at the top. One of my cousins studies at Presidency University, Kolkata. And he vouches for the Economics degree provided by the institute.

Taking into consideration various factors like fees, faculty, it is safe to say Pracidency University is the better alternative.

But if you are looking for placement after graduation, then joining Presdancyi will not be a good idea. Although their Economics dept. Is well-renowned, they don’t offer placements after graduation. You will have better placement opportunities at BITS Goa.

The preference is always given to BITS Goa as it belongs to the most reputed family of BITS institute. One of my friends has studied there and he loves the campus from all aspects.

The major reasons as to why BITS Goa is to be preferred are:

Scope

• CSE has more job opportunities in the market as compared to other branches of engineering. 
• If you even go for a core organization in the chemical field; the highest package is not more than 6 LPA. 
• In the case of CSE, if you are into coding a good package can be cracked.
• The highest package is close to 1.40 crore in BITS Goa. The average also stands at a decent place. 
• The prominent recruiters are - ADM Argo, Adobe, Bajaj Auto, Capgemini, CGI, Cisco, DE Shaw, EXL Services, Razor Pay, Reliance Jio, Freshworks and Goldman Sachs.

Faculty

• The computer science faculty cannot be compared with chemical engineering. 
• CS faculty is amazing having well-equipped knowledge in the field to deal with real competition. They all are efficient in imparting real practical skill set to their students in a proper manner.

Campus

• Goa is a place people look out for holidays, and having a college in Goa makes it even more fascinating and attractive. 
• Certain limitations are also imposed on students. You cannot leave the campus before 5 PM and you should return back by 11 PM. 

BITS Goa makes it a better option on all fronts whether its faculty, campus, or scope. Having this outlook will help you get to a good institute. The stream can be a point of interest but other factors also matter.

Admission at Birla Institute of Technology and Science, Goa, and Hyderabad is based on an entrance exam known as BITSAT. Admission is fully based on the results of such exam. College provides almost 100% of the placement. Physics, Chemistry, and Math are necessary subjects to have in the 12th board.

Admission in IIST Shibpur for IT is based on the results of JEE mains. And students are required to have physics, chemistry and math as primary subjects in 10+2. In IIST, IT has the highest number of placements.

Tabulated below is a brief overview of the institutes:

Admission process:

BITS Goa/Hyderabad

• The application process is fully online. The application fee is 3,150 INR for males and 2,650 for female applicants.
• Admissions are totally based on the results of the BITSAT entrance.
• Students are also required to complete the 12th board with at least 75%. Physics chemistry and math should be primary subjects in boards.

IIEST Shibpur

• Admission is based on the results of JEE Mains.Students are also required to complete 12th by 75%(For ST, SC and Physically challenged — 65% is required in 10+2).
• The whole application process is online. The application fee is 1500 INR.
• A total of60 seats are available for admission in IT from IIEST Shibpur.

Placements:

BITS Goa/Hyderabad

• This Institute has a really good placement rate, almost about 90 to 100%. Top recruiters are Amazon, Microsoft, Aditya Birla, Qualcomm, xilinx, and so on.
• The average salary provided by the colleges is 9to 10 lakhs.

IIEST Shibpur

• Companies like Google, Microsoft, Adobe, Direct, etc, visit campus each year for placements.
• The average salary offered to students is about 7 lakhs.This year the highest package was of about 40 lakhs offered by Google and Microsoft.

Both the Institutes are good at their level. IIEST-Shibpur is one of the oldest institutions and BITS Goa/Hyderabad is one of the young institutes in India. Facilities and infrastructure is really good of both the institutes and with this top companies visit campuses every year in search of placement.