How many active large commercial airplanes are there?
As mentioned this question is hard to answer very accurately without better access to data. It also depends on your definition of "large, commercial aircraft." Including all wide-body aircraft would be a large number of planes, and the totals are constantly changing as aircraft are delivered and retired.

This site will probably be the best resource for this kind of question. The data is probably not 100% accurate but it seems to be close. Other information is also available here.

I looked at the widebody models on the first linked site, sorting by status to find how many are in "Active" service. This does not include "Stored" aircraft which will be either put into service or not.

The models below total: 1026 cargo, 3825 passenger. This means about 21% of the 4851 widebodies are cargo aircraft, and widebodies constitute about 24% of the worldwide commercial aircraft fleet of 20000 aircraft.

747

There are 652 aircraft in service of 1500 delivered to date. Of these, about 289 are cargo aircraft. Excluding 28 aircraft operated by militaries or other organizations, this leaves about 335 passenger aircraft.

DC-10

There are 69 aircraft in service of 446 delivered. This does not include aircraft in military or firefighting services, or the aircraft used by Orbis.

A300

There are 255 active of 561 delivered. Excluding 6 in military or other service, this leaves 178 in cargo service and 71 in passenger service.

767

There are about 821 767 aircraft in service of 1062 delivered. About 165 are cargo versions, and excluding 16 in military or other service, this leaves 640 in passenger service.

A310

There are 105 active of 255 delivered. Excluding 17 in military or other service, this leaves 14 in cargo service and 74 in passenger service.

MD-11

There are 153 aircraft in service of 200 delivered. Only 3 of these are passenger aircraft. These are operated by KLM, and the last scheduled flight is October 25, 2014.

A330

There are 1069 active of 1112 delivered. Excluding 22 in military or other service, this leaves 27 in cargo service and 1020 in passenger service.

A340

There are 289 active of 377 delivered. Excluding 11 in military or other service, this leaves 278 aircraft in passenger service. If there are any versions used only for cargo, there aren't many.

777

There are about 1216 777 aircraft in service of 1230 delivered. About 134 are cargo versions, leaving about 1082 in passenger service.

A380

There are 139 delivered. There are no cargo variants and none out of service.

787

There are 183 delivered. There are no cargo variants and none out of service.