Counting Digit 5 Occurrences 0-1000

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In summary: This would give you the number of numbers between 0 and 999 that have a zero in them. For example, the number 955 would be counted because it is three digits and it doesn't have a 5 in it. But the number 559 would not be counted because it is a five digit number and it has a 5 in it. Thus, the total number of numbers between 0 and 999 that have a 5 in them is 9+9+1=18. In summary, the number of numbers between 0 and 1000 that have a digit 5 in it is 27.
  • #1
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Homework Statement


How many times does digit 5 occur in numbers fro 0-1000.

Homework Equations




The Attempt at a Solution


This is what i have done.
Total (1,2,3) digit numbers which have digit 5 occurring once in them are-:
[itex] 3.^9C_1.^9C_1 = 243 [/itex]
Total numbers with 5 occurring twice are [itex] ^9C_1 = 9 [/itex]
So, digit 5 occurs 9*2=18 times
Total Numbes with 5 occurring thrice= 1
So digit 5 occurs 3*1=3
Total times digit 5 occurs is 243+18+1=264

Is this correct, specially regarding repetitions or exclusions which i may have made?
 
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  • #2
I would have done this quite differently. first, there is xy5 in every 10 numbers and there are 100 sets of 10 in 1000: 100 such numbers. In addition, there is 5x 9 times (not counting 55) in every 100, and 10 hundreds in 100, so 90 more Finally, every number from 500 to 599 has a 5. leaving out those of the form 5x5 and 55x, that's an additional 81. That makes a total of 100+ 90+ 81= 271 numbers between 1 and 1000 that have at least one digit 5.
 
  • #3
"Total numbers with 5 occurring twice are [itex] ^9C_1 = 9 [/itex]
So, digit 5 occurs 9*2=18 times"
I think you missed a few numbers on this one...

55, 155, 255, 355, 455, 655, 755, 855, 955
That's 9 numbers with two 5's. But also:
505,515,525,535,545,565,575,585,595
And:
550, 551, 552, 553, 554, 556, 557, 558, 559

Thus, there are 3 locations that the non-5 can be, thus there are
[itex] ^9C_1 *3= 9 *3[/itex] numbers with two 5's,
Resulting in a total of 27*2 = 54 5's.

Other than that, I probably would have initially used HallsofIvy's approach as well. However, after reviewing your approach, I think I like it better. (With your approach, your solution would match HallofIvy's solution for the number of distinct numbers with a 5 in it, rather than the number of times "5" occurs:
[itex] 3.^9C_1.^9C_1 = 243 [/itex]
[itex] +3.^9C_1 = 27 [/itex]
[itex] +1[/itex]

But, since you're counting the 5's, you have to multiply that 2nd line by 2 and the 3rd line by 3. Nice.
 
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  • #4
It seems like it can be read two different ways. If you just want to count the number of 5s in the numbers between 0 and 999, then you may as well write the numbers with three digits, that is, from 000 to 999, and then it should be clear that the three digits are independent. This is easy.

More difficult is to count the numbers that have any 5 in them. To do this, you might try instead counting the number of three digit numbers that don't have any 5 in them, and subtract that from 1000.
 

What is the purpose of counting the occurrences of digit 5 in numbers 0-1000?

The purpose of counting the occurrences of digit 5 in numbers 0-1000 is to analyze the frequency and distribution of this specific digit within a given range. This can provide insights into patterns and probabilities within numbers, which can be useful in fields such as statistics and data analysis.

What is the most efficient method for counting the occurrences of digit 5 in numbers 0-1000?

The most efficient method for counting the occurrences of digit 5 in numbers 0-1000 is to use a loop that iterates through each number and checks if it contains the digit 5. This can be done in a variety of programming languages and is a simple and effective way to count occurrences.

What is the expected number of occurrences of digit 5 in numbers 0-1000?

The expected number of occurrences of digit 5 in numbers 0-1000 is 100, as there are 100 numbers that contain the digit 5 (5, 15, 25, ..., 995, 1005, 1015, ...). This assumes that numbers are counted inclusively (i.e. 0 and 1000 are both included).

What are some potential applications of counting digit 5 occurrences in numbers 0-1000?

Counting digit 5 occurrences in numbers 0-1000 can have various applications, such as analyzing lottery numbers, identifying trends in stock prices, or detecting patterns in numerical data. It can also be used in educational settings to teach mathematical concepts, such as probability and number patterns.

Is there a limit to the range of numbers that can be used when counting digit 5 occurrences?

No, there is no limit to the range of numbers that can be used when counting digit 5 occurrences. However, as the range increases, the number of occurrences may also increase, making it more challenging to analyze and draw meaningful conclusions. It is important to consider the purpose and practicality of counting digit 5 occurrences in a given range.

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