If a bicycle has two sprockets on the front and four sprockets on the back, how many different gear combinations would it have?

The earlier answer is correct but very few cyclists would ever use all the combinations of gears available from a multi-gear system. The real advantage in such systems, say a 21-speed with three pedal sprockets and seven rear-wheel sprockets is that it gives a greater range i.e. very high or very low gears in addition to medium ones. This means of course that steeper climbs can be attempted in low gear and faster speeds achieved in high gear when cruising. The other reason that the full range would seldom be used is that it is better for your bike if you keep your chain as straight as possible most of the time i.e. not strung between the inner sprocket at the pedal and the outer ones at the rear wheel or vice versa. Apart from the fact that such combinations are rarely necessary, a chain running too diagonally between the sprockets will cause a bad pattern of wear on both chain and sprocket teeth. So, even though 21 permutations may be available in theory, most cycling - and good care of the bike -requires no more than about five or six to be used in practice.