what four-digit number satisfies the following conditions? the sum of the digits is 6; the number is less than 1,200; none of the four digits are equal; and the tens digit is an odd number
To find the four-digit number that satisfies the given conditions, we can start by listing all the possible combinations of four different digits that add up to 6. Let's denote the digits as A, B, C, and D.
Possible combinations of these digits:
1) A + B + C + D = 6
2) A + B + D + C = 6
3) A + C + B + D = 6
4) A + C + D + B = 6
5) A + D + B + C = 6
6) A + D + C + B = 6
Next, we need to consider that the tens digit must be an odd number. This leaves us with the following possibilities for the tens digit: 1, 3, 5, 7, 9.
Now, let's check each possibility with the condition that the number must be less than 1,200:
1) A + B + C + D = 6
- For 1 as the tens digit, the thousands digit can only be 0 or 1. No valid combinations.
- For 3 as the tens digit, the thousands digit can only be 0, 1, 2, or 3. Possible combinations: 3201, 3102.
- For 5 as the tens digit, the thousands digit can only be 0, 1, 2, 3, 4, or 5. Possible combinations: 5201, 5102, 4103, 4203.
- For 7 as the tens digit, the thousands digit can only be 0, 1, 2, 3, 4, 5, 6, or 7. Possible combinations: 7201, 7102, 6103, 6203, 5302, 6301.
- For 9 as the tens digit, the thousands digit can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Possible combinations: 9201, 9102, 8103, 8203, 7302, 8301, 9300.
From these possibilities, we need to remove any combinations that have repeating digits. After removing those, we are left with the following four-digit numbers that satisfy all the conditions:
3201, 3102, 5201, 5102, 4103, 4203, 7201, 7102, 6103, 6203, 5302, 6301, 9201, 9102, 8103, 8203, 7302, 8301, 9300.
Therefore, there are 18 four-digit numbers that satisfy the given conditions.
To find a four-digit number that satisfies the given conditions, we can follow these steps:
Step 1: Start with the fact that the sum of the digits is 6. Let's consider the possible combinations of four different digits that add up to 6:
Combination 1: 1 + 2 + 2 + 1 = 6
Combination 2: 1 + 1 + 2 + 2 = 6
Combination 3: 3 + 2 + 1 + 0 = 6
Combination 4: 1 + 1 + 1 + 3 = 6
Step 2: Next, consider the condition that the tens digit is an odd number. Out of the four combinations listed above, only combination 1 has an odd number in the tens place (i.e., 2).
Step 3: Finally, check if the number is less than 1,200. The four-digit number 1221 is less than 1,200.
Therefore, the four-digit number that satisfies all the given conditions is 1221.