In organic chemistry terms, the concept of "isomerism" refers to how certain organic compounds--namely alkane hydrocarbons--can exist in many different arrangements, even with the same molecular formula. For example, pentane (C5H12) can exist in three unique configurations, even though it will always have five carbon and 12 hydrogen atoms. In 1875, a mathematician named Arthur Cayley came up with a mathematical formula to calculate the number of isomers. Unfortunately, this--and all other non-computerised methods developed since then--was disproved. You can, however, extrapolate the number of isomers of a given alkane from the "enumeration" sequence he created.
Consider all alkanes in the following terms: CnH(2n+2). Keeping this in mind, determine "n" for your particular compound. For example, if you have C5H12 (pentane), "n" = 5.
Consult Cayley's Sequence to determine how many isomers this "n" alkane has. For any alkane CnH(n+2), where "n" is any integer between 1 and 20, the following sequence represents the number of isomers possible: 1, 1, 1, 2, 3, 5, 9, 18, 35, 75, 159, 355, 802, 1858, 4347, 10359, 24894, 60523, 148284, 366319. In the case of pentane--which has five carbons and therefore represented by the fifth integer of the sequence--the answer is three.
Draw out your configurations on a piece of scratch paper to check your understanding. Pentane, for example, with its five carbons and 12 hydrogens, exists most basically as a straight chain of five carbons, with three hydrogens bonded to each of the "end" carbons and two to each of the three in the middle. Try your hand at the other two isomers of pentane, keeping in mind that each carbon must have exactly four other atoms bonded to it and that every configuration must account for 12 hydrogens. Check the "Tips" section when you've finished for the answer.
For alkanes whose "n" value exceeds 20, consult the online isomer calculator listed in Resources. In addition to a straight chain, pentane can also exist as a "cross," with one carbon in the middle and the other four carbons bonded to each of its electron pairs. Each of these "outer" carbons will have three hydrogens bonded to them. Its third and final isomer constitutes a chain of four carbons with the fifth bonded to either of the "middle" carbons in the chain. The end carbons and this "protruding" carbon will have three hydrogens each bonded to them, while the "middle" carbon in question will have only one hydrogen. The remaining carbon will have two hydrogens. In both of these cases, you can see that each carbon has four items--be they hydrogens or another carbon--bonded to it and each configuration accounts for 12 hydrogens.