gb7nash said:You need to define your rules a little more. Can we start from any dot/number? In the example you have, can you go from 9 to 8? Are only horizontal/vertical lines aloud? etc.
The number of combinations that can be made from a set of n items is 2^n. This means that for every item in the set, there are 2 possible outcomes - it either is or isn't included in the combination. Therefore, the total number of combinations is calculated by raising 2 to the power of n.
The formula for calculating combinations with repetition is n^r, where n is the number of items in the set and r is the number of items in each combination. This means that for every item in the set, there are n possible outcomes, and we multiply these possibilities by the number of items in each combination to get the total number of combinations.
The number of combinations decreases when some items are fixed in a set. This is because with fixed items, there are fewer options for the remaining items to be included in the combination. The formula for calculating this is n! / (n-r)! where n is the total number of items in the set and r is the number of items that are fixed.
The main difference between combinations and permutations is that combinations focus on the selection of items without regard to the order in which they are selected, while permutations consider the order in which the items are selected. Therefore, the number of combinations is typically smaller than the number of permutations for the same set of items. The formula for calculating permutations is n! / (n-r)!, where n is the total number of items in the set and r is the number of items in each permutation.
Understanding combinations can be useful in various real-life situations, such as when creating passwords, choosing lottery numbers, or arranging a menu for a restaurant. It can also be helpful in fields like genetics, where different combinations of genes can result in different traits or characteristics. In computer science, combinations are used in algorithms and data structures to efficiently store and retrieve data.