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The number of 4 digit numbers greater than 5000 can be formed out of the digits 3,4,5,6 and 7(no. digit is repeated). The number of such is
- a)72
- b)27
- c)70
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
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The number of 4 digit numbers greater than 5000 can be formed out of t...
To form a 4-digit number greater than 5000 using the digits 3, 4, 5, 6, and 7 without repetition, we need to consider the following:
- The first digit has to be either 5, 6, or 7.
- The remaining digits can be any of the remaining digits, i.e., 4 digits out of the 4 available.
For the first digit, there are 3 choices (5, 6, or 7).
Then, for the remaining three digits, we have 4 choices for the second digit, 3 choices for the third digit, and 2 choices for the fourth digit, since we cannot repeat any digit.
So, the total number of such 4-digit numbers is 3×4×3×2 =72.
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The number of 4 digit numbers greater than 5000 can be formed out of t...
Solution:
To form 4 digit numbers greater than 5000, we must start the number with digit 5 or 6 or 7. Therefore, we have three cases.
Case 1: Number starts with 5
In this case, we have 4 choices for the first digit (5,6,7,8) and 4 choices for the second digit (excluding the digit already used in the first position). For the third digit, there are 3 choices left and for the fourth digit, there are 2 choices left.
Total number of numbers = 4 x 4 x 3 x 2 = 96
Case 2: Number starts with 6
In this case, we have 3 choices for the first digit (6,7,8) and 3 choices for the second digit (excluding the digit already used in the first position). For the third digit, there are 2 choices left and for the fourth digit, there is only 1 choice left.
Total number of numbers = 3 x 3 x 2 x 1 = 18
Case 3: Number starts with 7
In this case, we have 2 choices for the first digit (7,8) and 3 choices for the second digit (excluding the digit already used in the first position). For the third digit, there is only 1 choice left and for the fourth digit, there is only 1 choice left.
Total number of numbers = 2 x 3 x 1 x 1 = 6
Therefore, the total number of 4 digit numbers greater than 5000 that can be formed out of the digits 3,4,5,6 and 7 is 96 + 18 + 6 = 120.
However, we must exclude the numbers that start with digit 8 (since we only want numbers greater than 5000). There are 4 such numbers: 8234, 8243, 8324, 8342.
Therefore, the final answer is 120 - 4 = 116.
Hence, option (A) 72 is not the correct answer.
To form 4 digit numbers greater than 5000, we must start the number with digit 5 or 6 or 7. Therefore, we have three cases.
Case 1: Number starts with 5
In this case, we have 4 choices for the first digit (5,6,7,8) and 4 choices for the second digit (excluding the digit already used in the first position). For the third digit, there are 3 choices left and for the fourth digit, there are 2 choices left.
Total number of numbers = 4 x 4 x 3 x 2 = 96
Case 2: Number starts with 6
In this case, we have 3 choices for the first digit (6,7,8) and 3 choices for the second digit (excluding the digit already used in the first position). For the third digit, there are 2 choices left and for the fourth digit, there is only 1 choice left.
Total number of numbers = 3 x 3 x 2 x 1 = 18
Case 3: Number starts with 7
In this case, we have 2 choices for the first digit (7,8) and 3 choices for the second digit (excluding the digit already used in the first position). For the third digit, there is only 1 choice left and for the fourth digit, there is only 1 choice left.
Total number of numbers = 2 x 3 x 1 x 1 = 6
Therefore, the total number of 4 digit numbers greater than 5000 that can be formed out of the digits 3,4,5,6 and 7 is 96 + 18 + 6 = 120.
However, we must exclude the numbers that start with digit 8 (since we only want numbers greater than 5000). There are 4 such numbers: 8234, 8243, 8324, 8342.
Therefore, the final answer is 120 - 4 = 116.
Hence, option (A) 72 is not the correct answer.
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The number of 4 digit numbers greater than 5000 can be formed out of the digits 3,4,5,6 and 7(no. digit is repeated). The number of such isa)72b)27c)70d)none of theseCorrect answer is option 'A'. Can you explain this answer?
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