In a small northern city lottery, you must select four numbers from nineteen different numbers to win the big prize. The numbers have to be in a particular order. How many 4-number selections are possible? In how many distinguishable ways can the following letters be arranged? A A A B B B B C C D D D D Number of arrangements =
The number of 4-number selections from 19 different numbers can be found using the concept of combinations. Since the order matters in this case, it is a permutation problem.
To find the number of permutations, we can calculate using the formula:
P(n, r) = n! / (n-r)!
Here, n is the total number of options (19), and r is the number of choices we need to make (4).
So, P(19, 4) = 19! / (19-4)!
= 19! / 15!
= (19 * 18 * 17 * 16 * 15!) / 15!
= 19 * 18 * 17 * 16
Therefore, there are 19 * 18 * 17 * 16 = 65,536 possible 4-number selections.
To find the number of distinguishable ways the given letters can be arranged, we can use the concept of permutations.
Since there are repeating letters (A, B, C, D), we need to consider their frequencies.
The given letters can be arranged in the following way:
A A A B B B B C C D D D D
The number of arrangements is given by the formula:
n! / (a! * b! * c! * ...)
Here, n is the total number of letters (13), and a, b, c, ... denote the frequencies of each letter.
So, the number of arrangements = 13! / (3! * 4! * 2! * 4!)
= (13 * 12 * 11 * 10!) / (3! * 2!)
= (13 * 12 * 11) / 3!
Therefore, the number of arrangements is (13 * 12 * 11) / 3! = 286.
To calculate the number of 4-number selections in the lottery, you need to use the concept of permutations.
Since you must select four numbers from nineteen different numbers, the total number of ways to select the first number is 19.
After selecting the first number, there are only 18 remaining numbers to choose from for the second number.
Similarly, for the third number, there are 17 remaining numbers, and for the fourth number, there are 16 remaining numbers.
To find the total number of 4-number selections, you multiply the number of choices for each position:
19 × 18 × 17 × 16 = 65,536.
Therefore, there are 65,536 possible 4-number selections in the lottery.
Next, let's calculate the number of distinguishable ways to arrange the given letters: A A A B B B B C C D D D D.
To find the number of arrangements, we need to consider the repeated letters.
The total number of letters is 4 A's, 4 B's, 2 C's, and 4 D's.
To find the number of arrangements, we need to calculate the number of permutations of these letters. We can use the formula for permutations with repetition.
The formula for permutations with repetition is:
n! / (n1! × n2! × n3! ...)
Where n is the total number of objects, and n1, n2, n3, etc., are the counts of each repeated object.
Using this formula, we can calculate the number of arrangements of the given letters:
Total objects (n) = 13 (4 A's + 4 B's + 2 C's + 4 D's)
Repeating objects' counts:
n1 = 4 (A's)
n2 = 4 (B's)
n3 = 2 (C's)
n4 = 4 (D's)
Plugging these values into the formula:
Number of arrangements = 13! / (4! × 4! × 2! × 4!)
Calculating this expression, we get:
Number of arrangements = 8,468,200.
Therefore, there are 8,468,200 distinguishable ways to arrange the given letters: A A A B B B B C C D D D D.