In fact both, the Custom Department and RBI are correct. Passengers visits for different purposes. So they are entitled to different amount of foreign exchange. Foreign exchange can only be purchased from authorised persons. So there may be any amount. However specifically for tourism only US$ 10000 can be taken in a calendar year.
There is a limit, though I am surprised by Mumbai Custom website wordings. In the same website if you scroll down you will see limits on hard currency you can carry.
Travellers are allowed to purchase foreign currency notes/coins only up to US$ 2000. Balance amount can be taken in the form of traveller’s cheque or banker’s draft. Exceptions to this are (a) travellers proceeding to Iraq and Libya can draw foreign exchange in the form of foreign currency notes and coins not exceeding US$ 5000 or its equivalent; (b) travellers proceeding to the Islamic Republic of Iran, Russian Federation and other Republics of Commonwealth of Independent States can draw entire foreign exchange released in form of foreign currency notes or coins.
As a Tourist, this is a total limit
In connection with private visits abroad, viz., for tourism purposes, etc., foreign exchange up to US$10,000, in any one calendar year may be obtained from an authorised dealer. The ceiling of US$10,000 is applicable in aggregate and foreign exchange may be obtained for one or more than one visits provided the aggregate foreign exchange availed of in one calendar year does not exceed the prescribed ceiling of US$10,000 {The facility was earlier called B.T.Q or F.T.S.}. This US$10,000 (BTQ) can be availed of by a person alongwith foreign exchange for travel abroad for any purpose, including for employment or immigration or studies. However, no foreign exchange is available for visit to Nepal and/or Bhutan for any purpose.
For Business Travel
Authorised dealers can release foreign exchange up to US$25,000 for a business trip to any country other than Nepal and Bhutan. Release of foreign exchange exceeding US$25,000 for a travel abroad (other than Nepal and Bhutan) for business purposes, irrespective of period of stay, requires prior permission from Reserve Bank. Visits in connection with attending of an international conference, seminar, specialised training, study tour, apprentice training, etc., are treated as business visits. Visit abroad for medical treatment and/or check up also falls within this category.
There is really no limit set on the amount of cash you can carry into the country.
The only difference is whether or not you need to declare it when you cross the border which according to EU Regulation 1809/2005 is 10000 Euro.
The only thing is that when you carry amounts of that size or larger Customs and various other agencies start to get curious on where this money came from.
Additional Info as per @Annoyed’s comment
From the site of the Customs office at Mumbai Airport
Export of foreign exchange/currency notes:
Indian residents going abroad are allowed to take with them foreign currency upto any amount so long as the same has been purchased from an authorized foreign exchange dealer as per norms.
Any person resident out of India is allowed to take out of India unspent foreign currency not exceeding the amount brought by him on his arrival in India provided that in the case of foreign currency notes exceeding US$ 5,000/- or its equivalent or foreign exchange in the form of currency notes, bank notes or traveller’s cheques exceeding US$ 10,000/- or its equivalent, the same was declared to the Customs authorities in Currency Declaration Form (CDF) at the time of his arrival in India.
So long story short. Still no limitation as long as the source can be proven legitimate.
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