![]() ![]() When all or some of the elements of a given collection of items are arranged in a definite arrangement or an order, or when the already existing arrangement of the same objects is rearranged into different orders, various permutations are created. The 1 stands for the choice of not selecting anything. Since not selecting an item or element (which ideally can be represented in numerical terms as 0!) is also a formal choice, 0! = 1. The factorial function exists for non-negative integers only. It is vital that we know the factorial function since combinatorics heavily relies on it. The factorial function is denoted by an exclamation mark.įor instance, 7 factorial is denoted as 7! and its mathematical value is 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040. The factorial of any positive integer is obtained by multiplying every positive integer lesser than the concerned number to it. ∴ According to the rule of sum, the number of possible ways in which only a dish can be chosen and bought = 3 + 2 = 5 ways ![]() There are 3 options for store A in case we chose it or there are 2 options for store B in case we chose it. We can either buy 1 of the 3 dishes from store A or 1 of the 2 dishes from store B. When we are to buy a single dish from either of the stores, we apply the rule of sum and figure out the total number of ways in which we can do it. Store A sells French fries, pizza and burger while store B sells waffle and cake. Given below are the dishes at two stores, A and B. Let us see an example where there are two factors. Note that the rule of sum can be extended to more than two factors as well. If there are ‘ m’ number of choices or ways for doing something and ‘ n’ number of choices or ways for doing another thing and they cannot be done together at the same time, then there are m + n ways of doing one of all those things. ∴ According to the rule of product, the number of possible ways to cross the town = 3 X 2 X 2 = 12 ways Rule of sum: Note that the whole deal will occur in stages, the first task being the selection of 1 of the 3 cafes, the second being the selection of 1 of the 2 banks and the third being the selection of 1 of the 2 libraries. Although the number is finite, it will take you a while to figure out the total number of ways in which it can be accomplished. This approach is laborious and time consuming. For example, one could enter the town, go to café C1, then to bank B1, and then go through library L1 and exit the town. ![]() To find the ways to cross this town and get to its end, you could manually start counting and framing routes randomly. Finally, the roads from the libraries converge into a path with the red dot on it, marking the end of the town. From the row of the 2 banks originates a common path to the final row of library buildings, L1 and L2. The path from the cafes leads to a row of 2 banks, B1 and B2. Then, we have a path to a row of 3 cafes, C1, C2 and C3. In the aerial view of the town given below, the green dot on the left-hand side marks the entry of the town. Let us see an example where there are 3 factors. Note that the rule of product can be extended to more than two factors as well. If a certain action can be performed in ‘ a’ number of ways and another, in ‘ b’ number of ways, then both these actions can be done in a x b number of ways. These concepts not only help us tell apart one set of things from another, but also make us grasp how the items of any single group can be arranged in numerous patterns amongst themselves.įundamental principle of counting: Rule of product: Permutation and combination employ these techniques and spare us the effort of manually enumerating the desired outcomes one by one. The branch of mathematics concerned with the various methods of counting is known as Combinatorics. To do this, we simply use certain counting techniques. ![]() The prime reason behind studying mathematics is to be able to count and to be able to arrive at answers. ![]()
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