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The total number of 3 digit numbers which have two or more consecutive digits identical is:
- a)171
- b)170
- c)90
- d)180
- e)None of these
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The total number of 3 digit numbers which have two or more consecutive...
Answer a)
In each set of 100 numbers, there are 10 numbers whose tens digit and unit digit are same. Again in the same set there are 10 numbers whose hundreds and tens digits are same. But one number in each set of 100 numbers whose Hundreds, Tens and Unit digit are same as 111, 222, 333, 444 etc
Hence, there are exactly (10 + 10 - 1) = 19 numbers in each set of 100 numbers. Further there are 9 such sets of numbers
Therefore such total numbers = 19 � 9 = 171
Alternatively,
9 � 10 � 10 - 9 � 9 � 9 = 900 - 729 = 171
Community Answer
The total number of 3 digit numbers which have two or more consecutive...
Total number of 3 digit numbers with two or more consecutive digits identical:
To find the total number of 3 digit numbers with two or more consecutive digits identical, we can break down the problem into different cases:
Case 1: Two consecutive digits are the same
In this case, we have 9 choices for the first digit (1-9), 1 choice for the second digit (since it has to be the same as the first digit), and 10 choices for the third digit (0-9). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 10 = 90.
Case 2: Three consecutive digits are the same
In this case, we have 9 choices for the first digit, 1 choice for the second digit (since it has to be the same as the first digit), and 1 choice for the third digit (since it has to be the same as the first and second digits). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 1 = 9.
Case 3: Four consecutive digits are the same
In this case, we have 9 choices for the first digit, 1 choice for the second digit (since it has to be the same as the first digit), and 1 choice for the third digit (since it has to be the same as the first and second digits). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 1 = 9.
Case 4: Five consecutive digits are the same
In this case, we have 9 choices for the first digit, 1 choice for the second digit (since it has to be the same as the first digit), and 1 choice for the third digit (since it has to be the same as the first and second digits). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 1 = 9.
Case 5: Six consecutive digits are the same
In this case, we have 9 choices for the first digit, 1 choice for the second digit (since it has to be the same as the first digit), and 1 choice for the third digit (since it has to be the same as the first and second digits). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 1 = 9.
Case 6: Seven consecutive digits are the same
In this case, we have 9 choices for the first digit, 1 choice for the second digit (since it has to be the same as the first digit), and 1 choice for the third digit (since it has to be the same as the first and second digits). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 1 = 9.
Case 7: Eight consecutive digits are the same
In this case, we have 9 choices for the first digit, 1 choice for the second digit (since it has to be the same as the first digit), and 1 choice for the third digit (since it has to be the same as the first and second digits). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 1 = 9.
Case 8: Nine
To find the total number of 3 digit numbers with two or more consecutive digits identical, we can break down the problem into different cases:
Case 1: Two consecutive digits are the same
In this case, we have 9 choices for the first digit (1-9), 1 choice for the second digit (since it has to be the same as the first digit), and 10 choices for the third digit (0-9). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 10 = 90.
Case 2: Three consecutive digits are the same
In this case, we have 9 choices for the first digit, 1 choice for the second digit (since it has to be the same as the first digit), and 1 choice for the third digit (since it has to be the same as the first and second digits). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 1 = 9.
Case 3: Four consecutive digits are the same
In this case, we have 9 choices for the first digit, 1 choice for the second digit (since it has to be the same as the first digit), and 1 choice for the third digit (since it has to be the same as the first and second digits). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 1 = 9.
Case 4: Five consecutive digits are the same
In this case, we have 9 choices for the first digit, 1 choice for the second digit (since it has to be the same as the first digit), and 1 choice for the third digit (since it has to be the same as the first and second digits). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 1 = 9.
Case 5: Six consecutive digits are the same
In this case, we have 9 choices for the first digit, 1 choice for the second digit (since it has to be the same as the first digit), and 1 choice for the third digit (since it has to be the same as the first and second digits). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 1 = 9.
Case 6: Seven consecutive digits are the same
In this case, we have 9 choices for the first digit, 1 choice for the second digit (since it has to be the same as the first digit), and 1 choice for the third digit (since it has to be the same as the first and second digits). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 1 = 9.
Case 7: Eight consecutive digits are the same
In this case, we have 9 choices for the first digit, 1 choice for the second digit (since it has to be the same as the first digit), and 1 choice for the third digit (since it has to be the same as the first and second digits). Hence, the total number of 3 digit numbers in this case is 9 * 1 * 1 = 9.
Case 8: Nine
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The total number of 3 digit numbers which have two or more consecutive digits identical is:a)171b)170c)90d)180e)None of theseCorrect answer is option 'A'. Can you explain this answer?
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