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The number of 4-digit numbers that can be made with the digits 1,2, 3, 4 and 5 in which at least two digits are identical, is
- a)45-5!
- b)505
- c)600
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The number of 4-digit numbers that can be made with the digits 1,2, 3,...
The number of numbers when repelition is allowed = 54.
The number of numbers when digits cannot be repeated = 5P5.
So, the required number of numbers = 54-5!
The number of numbers when digits cannot be repeated = 5P5.
So, the required number of numbers = 54-5!
Most Upvoted Answer
The number of 4-digit numbers that can be made with the digits 1,2, 3,...
The number of numbers when repelition is allowed = 54.
The number of numbers when digits cannot be repeated = 5P5.
So, the required number of numbers = 54-5!
The number of numbers when digits cannot be repeated = 5P5.
So, the required number of numbers = 54-5!
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Community Answer
The number of 4-digit numbers that can be made with the digits 1,2, 3,...
Understanding the Problem
To find the number of 4-digit numbers that can be formed with the digits 1, 2, 3, 4, and 5 where at least two digits are identical, we can approach this problem using complementary counting.
Step 1: Total 4-Digit Numbers
- The total number of 4-digit numbers with the digits 1 to 5 is calculated by allowing repetition.
- Each digit can be chosen independently, so:
- Total combinations = 5^4 = 625
Step 2: Counting Distinct 4-Digit Numbers
- We count the cases where all digits are distinct.
- We can choose any 4 digits from the available 5, which can be done in "5 choose 4" ways and then arrange those digits.
- Choose 4 digits: 5C4 = 5
- Arrange those 4 digits: 4! = 24
- Total distinct arrangements = 5 * 24 = 120
Step 3: At Least Two Identical Digits
- To find the number of 4-digit numbers with at least two identical digits, we subtract the number of distinct combinations from the total combinations.
- Total with at least two identical digits = Total combinations - Distinct combinations
- Calculation: 625 - 120 = 505
Final Answer
The number of 4-digit numbers that can be formed with the digits 1, 2, 3, 4, and 5 in which at least two digits are identical is 505. This confirms that the correct answer is option 'B'.
To find the number of 4-digit numbers that can be formed with the digits 1, 2, 3, 4, and 5 where at least two digits are identical, we can approach this problem using complementary counting.
Step 1: Total 4-Digit Numbers
- The total number of 4-digit numbers with the digits 1 to 5 is calculated by allowing repetition.
- Each digit can be chosen independently, so:
- Total combinations = 5^4 = 625
Step 2: Counting Distinct 4-Digit Numbers
- We count the cases where all digits are distinct.
- We can choose any 4 digits from the available 5, which can be done in "5 choose 4" ways and then arrange those digits.
- Choose 4 digits: 5C4 = 5
- Arrange those 4 digits: 4! = 24
- Total distinct arrangements = 5 * 24 = 120
Step 3: At Least Two Identical Digits
- To find the number of 4-digit numbers with at least two identical digits, we subtract the number of distinct combinations from the total combinations.
- Total with at least two identical digits = Total combinations - Distinct combinations
- Calculation: 625 - 120 = 505
Final Answer
The number of 4-digit numbers that can be formed with the digits 1, 2, 3, 4, and 5 in which at least two digits are identical is 505. This confirms that the correct answer is option 'B'.
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The number of 4-digit numbers that can be made with the digits 1,2, 3, 4 and 5 in which at least two digits are identical, isa)45-5!b)505c)600d)none of theseCorrect answer is option 'B'. Can you explain this answer?
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