It is possible to have 32bit components on a 64bit system. The IE is the classic case. Look into your All Programs and you will find 2 IEs - 64bit and 32bit.
My Computer
Computer Manufacturer/Model Number
HP, Dell, Gateway, Toshiba - 4 laptops and 2 desktops
Wouldn't that make my programs more slow? I mean my system is 64-bit and there's a reason why I installed the 64-bit version. I heard it makes programs run faster if you have a RAM with 4GB+.
Edit:
Look at the task manager. Almost all programs run under 32-bit, even those I installed as 64-bit. Everything was fine before this update!
Microsoft recommends using the 32 bit version of Office 2013 and this is installed by default, even on a 64 bit OS. This is to avoid compatibility problems with third party add-ons, most of which are 32 bit only. The 64 bit version is provided for people who need the advantages it provides. 64 bit Excel and Project 2013 provide major benefits if you are using very large datasets.
64 bit Windows was never about performance, that being a relatively minor bonus in some situations. For most application types 32 bit and 64 bit applications will have similar performance. 64 bit systems were developed to provide a much larger address space for native 64 bit applications. This is complex and I will not describe it here. The ability to address more than 4 GB RAM is another bonus, but not what it was designed for.
During a Windows update you will receive 32 bit updates as well as 64 bit. The 32 bit subsystem is an almost complete installation of a 32 bit OS which requires updates as well.
Edit: Internet Explorer is just one example why both 64 bit and 32 bit versions are supplied. The 64 bit version has some advantages but the big downside is compatibility with third party extensions. Most of these are 32 bit only and will not work with the 64 bit version.