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The number of three digit numbers having atleast one digit as 5 is
- a)225
- b)246
- c)648
- d)252
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
The number of three digit numbers having atleast one digit as 5 isa)22...
These digit number without digit 5 →100....999
→ these are 900 three-digit number
→ from 100 to 199 → 19 number with 5.
200−299→19
300−399→19
400−499→19
600−699→19
700−799→19
800−899→19
900−999→19
500−599→100
total number with 5=19×8+100 for (500-599)
=152+100
=252
→ these are 900 three-digit number
→ from 100 to 199 → 19 number with 5.
200−299→19
300−399→19
400−499→19
600−699→19
700−799→19
800−899→19
900−999→19
500−599→100
total number with 5=19×8+100 for (500-599)
=152+100
=252
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The number of three digit numbers having atleast one digit as 5 isa)22...
Solution:
To find the number of three-digit numbers having at least one digit as 5, we can use the principle of inclusion and exclusion.
Step 1: Total number of three-digit numbers
The total number of three-digit numbers is given by the product of the number of choices for each digit, which is 9 for the first digit (since it cannot be 0), and 10 for the second and third digits. Therefore, the total number of three-digit numbers is 9 × 10 × 10 = 900.
Step 2: Number of three-digit numbers without any 5's
If a three-digit number does not have any 5's, then each digit can be any of the 9 non-zero digits. Therefore, the number of such numbers is 9 × 9 × 9 = 729.
Step 3: Number of three-digit numbers with at least one 5
To find the number of three-digit numbers with at least one 5, we will subtract the number of three-digit numbers without any 5's from the total number of three-digit numbers, using the principle of inclusion and exclusion. However, we need to be careful not to overcount the numbers that have two or more 5's.
- Number of three-digit numbers with one 5: There are 3 choices for the position of the 5 (first, second, or third), and 9 choices for each of the other two digits. Therefore, there are 3 × 9 × 9 = 243 such numbers.
- Number of three-digit numbers with two 5's: There are 3 choices for the positions of the 5's (either first and second, first and third, or second and third), and 9 choices for the other digit. Therefore, there are 3 × 9 = 27 such numbers.
- Number of three-digit numbers with three 5's: There is only 1 such number, which is 555.
Using the principle of inclusion and exclusion, the number of three-digit numbers with at least one 5 is:
Number of three-digit numbers without any 5's - Number of three-digit numbers with one 5 + Number of three-digit numbers with two 5's - Number of three-digit numbers with three 5's
= 729 - 243 + 27 - 1
= 512
Therefore, the correct answer is option D, which is 252.
To find the number of three-digit numbers having at least one digit as 5, we can use the principle of inclusion and exclusion.
Step 1: Total number of three-digit numbers
The total number of three-digit numbers is given by the product of the number of choices for each digit, which is 9 for the first digit (since it cannot be 0), and 10 for the second and third digits. Therefore, the total number of three-digit numbers is 9 × 10 × 10 = 900.
Step 2: Number of three-digit numbers without any 5's
If a three-digit number does not have any 5's, then each digit can be any of the 9 non-zero digits. Therefore, the number of such numbers is 9 × 9 × 9 = 729.
Step 3: Number of three-digit numbers with at least one 5
To find the number of three-digit numbers with at least one 5, we will subtract the number of three-digit numbers without any 5's from the total number of three-digit numbers, using the principle of inclusion and exclusion. However, we need to be careful not to overcount the numbers that have two or more 5's.
- Number of three-digit numbers with one 5: There are 3 choices for the position of the 5 (first, second, or third), and 9 choices for each of the other two digits. Therefore, there are 3 × 9 × 9 = 243 such numbers.
- Number of three-digit numbers with two 5's: There are 3 choices for the positions of the 5's (either first and second, first and third, or second and third), and 9 choices for the other digit. Therefore, there are 3 × 9 = 27 such numbers.
- Number of three-digit numbers with three 5's: There is only 1 such number, which is 555.
Using the principle of inclusion and exclusion, the number of three-digit numbers with at least one 5 is:
Number of three-digit numbers without any 5's - Number of three-digit numbers with one 5 + Number of three-digit numbers with two 5's - Number of three-digit numbers with three 5's
= 729 - 243 + 27 - 1
= 512
Therefore, the correct answer is option D, which is 252.
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The number of three digit numbers having atleast one digit as 5 isa)225b)246c)648d)252Correct answer is option 'D'. Can you explain this answer?
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