Nine cards are numbered 1, 2, 2, 3, 3, 4, 6, 6, 6.Three of the nine cards are chosen and placed in a line, making a 3-digit number. Find how
many different numbers can be made in this way if the number is between 200 and 300.
There are three digits in your number ___ ___ ___
We sometimes refer to them as "slots"
The first slot is where the hundreds is placed.
For your number to be between 200 and 300 only the 2's could go in that slot
_ 2_ , ___, ___
Then there are 8 other numbers that could go in the next slot (since one of the two 2's is used up in the first slot), then 7 numbers left for the next slot
2 x8 x 7
be sure to adjust for duplicate digits, since the equal digits are indistinguishable.
To determine the number of different numbers that can be made, we need to consider the possible combinations of the three chosen cards.
Step 1: Identify the available cards:
The available cards are 1, 2, 2, 3, 3, 4, 6, 6, 6.
Step 2: Determine the restrictions:
The number needs to be between 200 and 300, so the first digit must be 2.
Step 3: Identify the possible values for the second and third digits:
The possible values for the second digit are 0, 1, 2, 3, 4, 6, and the possible values for the third digit are 0, 1, 2, 3, 4, 6.
Step 4: Calculate the number of possible combinations:
For the second digit, we have 7 options (excluding 6 since it cannot be the second digit).
For the third digit, we have 6 options (excluding 6 and the digit chosen for the second digit).
Therefore, the number of different numbers that can be made is 7 * 6 = 42.
To find out how many different numbers can be made in this way, we need to consider the conditions given.
First, let's count how many times each digit appears in the set of cards:
- The digit 1 appears once.
- The digit 2 appears twice.
- The digit 3 appears twice.
- The digit 4 appears once.
- The digit 6 appears three times.
Since we need to form a 3-digit number between 200 and 300, there are a few conditions we need to consider:
1. The first digit must be 2.
2. The remaining two digits can be chosen from the set of remaining cards.
Let's break down the cases to find the answer:
1. Case 1: The first digit is 2.
- We have used one 2 from the set of cards.
- We have four cards left: 2, 3, 3, 4.
- For the second digit, we can choose one of the remaining four cards.
- For the third digit, we can choose one of the remaining three cards.
- The number of possible numbers for this case = 4 * 3 = 12.
2. Case 2: The first digit is 2.
- We have used one 2 from the set of cards.
- We have four cards left: 2, 3, 3, 6.
- For the second digit, we can choose one of the remaining four cards.
- For the third digit, we can choose one of the remaining three cards.
- The number of possible numbers for this case = 4 * 3 = 12.
3. Case 3: The first digit is 2.
- We have used one 2 from the set of cards.
- We have four cards left: 2, 3, 4, 6.
- For the second digit, we can choose one of the remaining four cards.
- For the third digit, we can choose one of the remaining three cards.
- The number of possible numbers for this case = 4 * 3 = 12.
Therefore, the total number of different numbers that can be made in this way, between 200 and 300, is:
12 (Case 1) + 12 (Case 2) + 12 (Case 3) = 36.
Hence, there are 36 different numbers that can be made in this way if the number is between 200 and 300.