How many numbers greater than 300 000 are there using only the digits 1, 1, 1, 2, 2, 3?
clearly the first digit must be the 3, so there are only the other 5 digits to arrange.
we have 3 alike of one kind and 2 alike of another,
so the number of ways = 1(5!)/(3!2!) = 10
Well, let's see. Among the digits 1, 1, 1, 2, 2, and 3, we can arrange them to form 6 distinct numbers: 111223, 111232, 111322, 112132, 112213, and 113122.
Now, if we want numbers greater than 300,000, we have to consider the placement of the digit 3. Since the digit 3 is the highest digit among the given digits, it must go in the leftmost position.
Therefore, there is only one number greater than 300,000 using the given digits, which is 321122.
So, we have a grand total of 1 number greater than 300,000.
To find the number of numbers greater than 300,000 using only the digits 1, 1, 1, 2, 2, and 3, we can use a combination of combinatorics and basic counting principles.
Let's break down the steps:
Step 1: Count the number of ways to arrange the given digits.
We have a total of 6 digits: 1, 1, 1, 2, 2, and 3.
The total number of arrangements is calculated using the permutation formula:
nPr = n! / (n - r)!
In this case, n = 6 (total number of digits) and r = 6 (all digits).
Number of arrangements = 6! / (6 - 6)! = 6! / 0! = 6! = 720
Step 2: Count the number of ways to arrange the digits that are less than or equal to 300,000.
To determine the number of ways to arrange the digits that are less than or equal to 300,000, we need to consider the restrictions imposed by the number itself.
Since the number must be greater than 300,000, the leftmost digit (hundred thousands place) can only be 3. The remaining places can be filled with any arrangement of the digits 1, 1, 1, 2, and 2.
We can calculate the number of arrangements for the remaining places (5 places) using the same permutation formula as in Step 1:
Number of arrangements = 5! / (5 - 5)! = 5! / 0! = 5! = 120
Step 3: Calculate the number of numbers greater than 300,000.
To find the number of numbers greater than 300,000 using only the digits 1, 1, 1, 2, 2, and 3, we subtract the number of arrangements from Step 2 from the total number of arrangements in Step 1:
Number of numbers > 300,000 = Total number of arrangements - Number of arrangements ≤ 300,000
Number of numbers > 300,000 = 720 - 120 = 600
Therefore, there are 600 numbers greater than 300,000 using only the digits 1, 1, 1, 2, 2, and 3.
To find the number of numbers greater than 300,000 using the digits 1, 1, 1, 2, 2, and 3, we need to consider the possible arrangements of these digits.
First, let's count the number of arrangements of these digits without any restrictions. We have 6 digits in total, so there are 6! (6 factorial) possible arrangements.
However, we need to consider that the first digit must be greater than 2 for numbers greater than 300,000. Since we have three 1s, two 2s, and one 3, there are a few possibilities for the first digit:
1. If the first digit is 3, then we have five remaining digits to arrange: 1, 1, 1, 2, 2. The number of arrangements is 5!/2!2! = 5!/(2!2!) = 5!/(4) = 5*4*3*2*1 = 120.
2. If the first digit is 2, then we have three remaining digits to arrange: 1, 1, 1. The number of arrangements is 3!/3! = 1.
3. If the first digit is 1, then we have two remaining digits to arrange: 1, 1. The number of arrangements is 2!/2! = 1.
Therefore, there are a total of 120 + 1 + 1 = 122 possible arrangements.
However, we also need to consider that some of these arrangements may result in the same number. For example, if we arrange the digits as 3, 1, 2, 1, 1, 2, we would get the same number as if we arranged them as 3, 2, 1, 1, 1, 2. To account for this, we need to divide the total number of arrangements by the number of arrangements for each repeated digit.
In this case, we have three 1s and two 2s, so we need to divide by 3! and 2!. Therefore, the final number of numbers greater than 300,000 is:
122 / (3! * 2!) = 122 / (6 * 2) = 122 / 12 = 10.1667.
Since we cannot have a fraction of a number, the final answer is 10.
So, there are 10 numbers greater than 300,000 using only the digits 1, 1, 1, 2, 2, and 3.