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The number of 3- digit numbers containin...

The number of 3- digit numbers containing the digit 7 exactly once :

A

225

B

220

C

200

D

180

Text Solution

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The correct Answer is:
To find the number of 3-digit numbers containing the digit 7 exactly once, we will consider three different cases based on the position of the digit 7. ### Step-by-Step Solution: **Case 1: The digit 7 is in the last position.** - The 3-digit number can be represented as _ _ 7. - The first digit (hundreds place) cannot be 0 (it must be a digit from 1 to 9). Therefore, we have 8 choices for the first digit (1 to 9, excluding 7). - The second digit (tens place) can be any digit from 0 to 9, but it cannot be 7 (to ensure 7 appears exactly once). Thus, we have 9 choices for the second digit (0 to 9, excluding 7). - The total combinations for this case = 8 (choices for the first digit) × 9 (choices for the second digit) = 72. **Case 2: The digit 7 is in the middle position.** - The 3-digit number can be represented as _ 7 _. - The first digit (hundreds place) again cannot be 0, so we have 8 choices (1 to 9, excluding 7). - The last digit (units place) can be any digit from 0 to 9, but it cannot be 7. Therefore, we have 9 choices for the last digit (0 to 9, excluding 7). - The total combinations for this case = 8 (choices for the first digit) × 9 (choices for the last digit) = 72. **Case 3: The digit 7 is in the first position.** - The 3-digit number can be represented as 7 _ _. - The first digit is fixed as 7, so we need to choose the second and third digits. - The second digit (tens place) can be any digit from 0 to 9, giving us 10 choices. - The third digit (units place) can also be any digit from 0 to 9, giving us 10 choices. - The total combinations for this case = 10 (choices for the second digit) × 10 (choices for the third digit) = 100. ### Total Combinations: Now, we sum the total combinations from all three cases: - Case 1: 72 - Case 2: 72 - Case 3: 100 Total = 72 + 72 + 100 = 244. Thus, the total number of 3-digit numbers containing the digit 7 exactly once is **244**.
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