If you put all 720 permutations of the digits 1 through 6 in numerical order from least to greatest (ex. 123456, 123465 etc.) What would the 409 permutation be? How do I do it?
Each permutaion of five numbers, following the first, has 120 possibilities. Therefore there are 360 numbers begining with 1, 2, or 3, and you are looking for the 49th number beginning with 4.
4! = 24 numbers will begin with each two-number combination. 361 through 384 will begin with 41. 385 through 408 will begin with 42. Therefore your number will be the first (smallest) number begining with 43.
What would that be?
431256
To find the 409th permutation of the digits 1 through 6 in numerical order, you can follow these steps:
Step 1: Find the first digit:
To determine the first digit of the 409th permutation, we need to divide 409 by the number of permutations of the remaining digits. In this case, we have 5 digits left, so there are 5! (5 factorial) permutations, which is equal to 120. Divide 409 by 120:
409 ÷ 120 = 3 remainder 49
Step 2: Find the second digit:
Now, take the remainder from step 1 (49) and divide it by the number of permutations of the remaining digits. Since we have 4 digits left, there are 4! (4 factorial) permutations, which is equal to 24. Divide 49 by 24:
49 ÷ 24 = 2 remainder 1
Step 3: Find the third digit:
Repeat the process with the new remainder from step 2 (1) and the number of permutations of the remaining digits. Since we have 3 digits left, there are 3! (3 factorial) permutations, which is equal to 6. Divide 1 by 6:
1 ÷ 6 = 0 remainder 1
Step 4: Find the fourth digit:
Again, use the new remainder from step 3 (1) and divide it by the number of permutations of the remaining digits (2! = 2). Divide 1 by 2:
1 ÷ 2 = 0 remainder 1
Step 5: Find the fifth digit:
Finally, divide the remainder from step 4 (1) by the number of remaining permutations (1! = 1). Divide 1 by 1:
1 ÷ 1 = 1
Step 6: Determine the permutation:
Now that we have determined the digits in the correct order, we can arrange them. Based on our findings from the previous steps, the digits for the 409th permutation will be 3, 2, 1, 1, 1, 6.
Therefore, the 409th permutation of the digits 1 through 6 in numerical order would be 321116.
To find the 409th permutation when arranging the digits 1 through 6 in numerical order, we can break down the problem into smaller steps:
Step 1: Determine the first digit.
Since the permutations are in numerical order, we can observe that the first digit remains the same for every 120 permutations. Therefore, we divide 409 by 120, which results in a quotient of 3 with a remainder of 49. This quotient shows us that the first digit of the 409th permutation is 4.
Step 2: Determine the second digit.
We need to find the 409th permutation within the set of permutations that start with the digit 4. To do this, we repeat the previous step with the remaining digits (1, 2, 3, 5, 6). We divide 49 by 24, and this time the quotient is 2 with a remainder of 1. This means that the second digit is 2.
Step 3: Determine the third digit.
Again, we find the 409th permutation within the set of permutations that start with the digits 42. We divide 1 by 6, and this time the quotient is 0 with a remainder of 1. Therefore, the third digit is 1.
Step 4: Determine the fourth digit.
Next, we find the 409th permutation within the set of permutations that start with the digits 421. We divide 1 by 2, resulting in a quotient of 0 with a remainder of 1. Hence, the fourth digit is 1.
Step 5: Determine the fifth digit.
We find the 409th permutation within the set of permutations that start with the digits 4211. We divide 1 by 1, yielding a quotient of 1 with no remainder. Consequently, the fifth digit is 6.
Step 6: Determine the sixth and final digit.
The remaining digit not used so far is 3, making it the last digit of the 409th permutation.
Putting it all together, the 409th permutation of the digits 1 through 6 in numerical order from least to greatest is 421116.