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. Something that we have to take into account is that the order of the elements is important, for example, if we have to order 3 elements, in the permutation is not only taken into account that 3 of the elements coincide, it will matter the order in which they were selected, where each selection is a different permutation.Īs we said in the definition of permutation, this is useful to define of how many ways we can classify or order a set of elements in a smaller set formed by elements of the major set, when we talk about a smaller set we are referring to extract elements from the set to form another one.The prime reason behind studying mathematics is to be able to count and to be able to arrive at answers. The repetition of elements is not something allowed in the permutation, this means that an element cannot be selected twice, something we can do with the combination. for example, if we have a set with 20 elements, the permutation would allows us to find the number of ways we can select a determined number of elements. The permutation is a mathematical method used in statistic where we can define of how many different ways we can select some elements from a set.
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