The fifth combinatorial technique that Hēmacandra describes is called *saṅkhyā*, which simply means “number”: it gives the number of possible combinations of light and heavy syllables for a verse of *k* positions. This is the simplest of all of the techniques. For *samavr̥tta* meters, which have an equivalent number of positions in each line, the solution is simply 2^{k}. Hence, as we have already seen, for *k* = 4, the number of possible combinations is 2^{4} = 16. For meters of the *śakvarī* class, with fourteen syllables per line, the number of possible combinations is 2^{14} = 16,384.

In case you want to find the number of possibilities *up to* a certain value of *k* (e.g., *k* = 1, 2, 3, 4 … 14), you can multiply the highest value of *k* by 2 and subtract 2 from the product. Hence, the total number of possibilities for *k* = 1, 2, 3, 4 … 14 is 2(2^{14}) -2 = 32,766.

For *ardhasamavr̥tta* meters, we have to modify the procedure somewhat. There should be a *greater* number of such meters: in *samavr̥tta* meters, all of the lines must be identical, but in *ardhasamavr̥tta* meters, only the first and third, and second and fourth, lines are identical to each other. Hence we start by raising the number of possibilities to the power of 2. For *k* = 4, this gives us (2^{4})^{2} = 256. But this number also includes patterns where all four lines are identical, i.e., the *samavr̥tta* forms. We subtract the total number of these *samavr̥tta* forms, which in this case is 2^{4} = 16. There are thus 240 *ardhasamavr̥tta* forms for *k* = 4. Hence the general formula is (2^{k})^{2} – 2^{k}.

For *mātrā*-based meters, we have to generalize the procedure a little. In *sama-* and *ardhasamavr̥tta* meters, there can only be 2 possibilities for any given position, namely, light or heavy. This is not the case in *mātrā*-based meters. We must therefore multiply the number of possibilities at each position (effectively: each group, or *gaṇa*, of *mātrā*). Ignoring for the moment the requirements of word-break, the *gāthā* has the following possibilities at each group:

First line: | 1^{st} gaṇa |
2^{nd} gaṇa |
3^{rd} gaṇa |
4^{th} gaṇa |
5^{th} gaṇa |
6^{th} gaṇa |
7^{th} gaṇa |
8^{th} gaṇa |
Product |
---|---|---|---|---|---|---|---|---|---|

Possibilities: | 4 | 5 | 4 | 5 | 4 | 2 | 4 | 1 | 12,800 |

Second line: | 1^{st} gaṇa |
2^{nd} gaṇa |
3^{rd} gaṇa |
4^{th} gaṇa |
5^{th} gaṇa |
6^{th} gaṇa |
7^{th} gaṇa |
8^{th} gaṇa |
Product |

Possibilities: | 4 | 5 | 4 | 5 | 4 | 1 | 4 | 1 | 6,400 |

Hence the number of possibilities is 12,800 × 6,400 = 81,920,000.

Finally, one might be interested, in principle, in how many possible combinations of light and heavy syllables can be accommodated within a unit (*gaṇa*) that contains a specified number of *mātrā*s. We know, for example, that four *mātrā*s can be realized by the following **five** patterns: ऽऽ, ।।ऽ, ।ऽ।, ऽ।।, ।।।।. But Hēmacandra tells us that the number of possibilities is a Fibonacci number. (Hēmacandra wrote about 50 years before Fibonacci, and the principle was known to earlier authors who were concerned with Prakrit metrics.) Thus:

Number of mātrās: |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

Number of possible syllabic realizations: | 0 + 1 = 1 | 1 + 1 = 2 | 2 + 1 = 3 | 2 + 3 = 5 | 3 + 5 = 8 | 5 + 8 = 13 | 8 + 13 = 21 | 13 + 21 = 34 |