![]() ![]() So we don't care about the order in which they were chosen, only which 3 pots were picked. Once the pots are chosen and gift wrapped, the recipient won't know (or care) which was selected first, which was selected second, etc. For example, we want to know how many ways we can choose 3 of the 5 flower pots to give as a gift. Combinations are used when (a) the choices for each step come from the same group, (b) without replacement, but (c) the order does not matter. The answer would be P(5,3) = 60.įinally, Combinations allows us to consider the selections without regard to the order. For example, we want to know how many ways we can arrange 3 of the 5 flower pots on the window sill. The flexibility comes in that we don't have to arrange all of the items in the group. It is used when (a) the choices for each step come from the same group, (b) without replacement, and (c) the order matters. Permutations are a more versatile variation of the factorial principle. The number of arrangements would be 5! = 5*4*3*2*1 = 120. For example, we want to know how many ways we can arrange five different flower pots on a window sill. Factorials are used when the choices for each step come from the same group (or set of options) without replacement, and we want to arrange (order) the entire group. For example, the first step could be picking 1 of 5 flower pots and the second step could be picking 1 of 3 types of flowers.Ī special form of the FCP is the factorial. The options for each step do not have to be related to each other. If you have n1 choices for the first step, n2 choices for the second step, n3 choices for the third step, etc. The most basic form is the Fundamental Counting Principle (FCP). Therefore the probability of winning the lottery is 1/13983816 = 0.000 000 071 5 (3sf), which is about a 1 in 14 million chance.The counting methods (FCP, factorials, permutations, and combinations) are used whenever you have a multi-step process with multiple options at each step. The number of ways of choosing 6 numbers from 49 is 49C 6 = 13 983 816. What is the probability of winning the National Lottery? You win if the 6 balls you pick match the six balls selected by the machine. In the National Lottery, 6 numbers are chosen from 49. The above facts can be used to help solve problems in probability. There are therefore 720 different ways of picking the top three goals. Since the order is important, it is the permutation formula which we use. In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. The number of ordered arrangements of r objects taken from n unlike objects is: How many different ways are there of selecting the three balls? The number of ordered arrangements of r objects taken from n unlike objects is: n P r n. There are 10 balls in a bag numbered from 1 to 10. The number of ways of selecting r objects from n unlike objects is: Therefore, the total number of ways is ½ (10-1)! = 181 440 How many different ways can they be seated?Īnti-clockwise and clockwise arrangements are the same. When clockwise and anti-clockwise arrangements are the same, the number of ways is ½ (n – 1)! The number of ways of arranging n unlike objects in a ring when clockwise and anticlockwise arrangements are different is (n – 1)! There are 3 S’s, 2 I’s and 3 T’s in this word, therefore, the number of ways of arranging the letters are: In how many ways can the letters in the word: STATISTICS be arranged? The number of ways of arranging n objects, of which p of one type are alike, q of a second type are alike, r of a third type are alike, etc is: The total number of possible arrangements is therefore 4 × 3 × 2 × 1 = 4! The third space can be filled by any of the 2 remaining letters and the final space must be filled by the one remaining letter. The second space can be filled by any of the remaining 3 letters. The first space can be filled by any one of the four letters. This is because there are four spaces to be filled: _, _, _, _ ![]() How many different ways can the letters P, Q, R, S be arranged? ![]() The number of ways of arranging n unlike objects in a line is n! (pronounced ‘n factorial’). This section covers permutations and combinations.
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