Find How many different numbers can be made by arranging all nine digits of the number
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hello friends so the question is find the total number of 9 digit number that can be formed using the digits 2233558888 so that or digit occupied even places so here how many numbers we have total 9 so 1 2 3 4 5 6 7 8 and 9 so how many odd numbers we have old and even numbers 5 and given is odd numbers ok you by the even places which wala even places this this this and this 2 4 6 and 8 so and the sport places even number should come so in this four places how many times can we read odd numbers we know that 3 + 2 X and 5 is also two times so what are the ways of arranging we get perfect Oriya divided by 2 into 2 factorial so we arrange odd numbers so 5 places which are left for even numbers are also we know that again we lock there are two tools and 38220 and 38 so we get ways of arranging even numbers we get 5 factorial divided by 2 factorial into 3 factorial so if we simplify for factorial divided by 2 into 2 factorial in 22 factorial into 5 factorial divided by 2 factorial into 3 factorial so early in these ways we can rearrange so by simplifying get 6 into 10 which is 60 days total 9 digit numbers can be formed in 60 days so what we did is we knew that odd numbers will come in even places so on even four places we rearranged odd numbers so 60 days will be our answer thank you This section covers permutations and combinations. Arranging Objects The number of ways of arranging n unlike objects in a line is n! (pronounced ‘n factorial’). n! = n × (n – 1) × (n – 2) ×…× 3 × 2 × 1 Example How many different ways can the letters P, Q, R, S be arranged? The answer is 4! = 24. This is because there are four spaces to be filled: _, _, _, _ The first space can be filled by any one of the four letters. The second space can be filled by any of the remaining 3 letters. 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 total number of possible arrangements is therefore 4 × 3 × 2 × 1 = 4!
n!
. Example In how many ways can the letters in the word: STATISTICS be arranged? 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: 10!=50 400 Rings and Roundabouts
When clockwise and anti-clockwise arrangements are the same, the number of ways is ½ (n – 1)! Example Ten people go to a party. How many different ways can they be seated? Anti-clockwise and clockwise arrangements are the same. Therefore, the total number of ways is ½ (10-1)! = 181 440 Combinations The number of ways of selecting r objects from n unlike objects is: Example There are 10 balls in a bag numbered from 1 to 10. Three balls are selected at random. How many different ways are there of selecting the three balls? 10C3 =10!=10 × 9 × 8= 120 Permutations A permutation is an ordered arrangement.
nPr = n! . Example In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. Since the order is important, it is the permutation formula which we use. 10P3 =10! = 720 There are therefore 720 different ways of picking the top three goals. Probability The above facts can be used to help solve problems in probability. Example In the National Lottery, 6 numbers are chosen from 49. You win if the 6 balls you pick match the six balls selected by the machine. What is the probability of winning the National Lottery? The number of ways of choosing 6 numbers from 49 is 49C6 = 13 983 816 . 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. |
