Combinatorics (Shai Simonson)

Question

How many n length base 10 numbers are there with at least 3 zeros?

Comments

When different number is asked :[9 * 10^n-1] - [9ⁿ + 9^n-1.^n-1c₁ + ^n-1c₂ * 9^n-2 ]

When no condition(different) on number :[10ⁿ] - [9ⁿ + 9^n-1.ⁿc₁ + ⁿc₂ * 9^n-2 ]

Whats the answer given .

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Atleast 3 zeroes can also be interpreted as Atmost 2 zeroes so we can have total number of n length base 10 numbers - the n length numbers having no zeros - the n length numbers having 1 zero - numbers having 2 zeroes

so the answer would be 10^n - (9^n - nc1*9*10^(n-1) -nc2*(9^2)*10^(n-2))   

Correct me if i am wrong

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if you take 1st place 0 then it becomes 9 digit number right.

nc2*(9^2)*10^(n-2) since you already choose 2 digit be 0 then 9² will be 1 only

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10ⁿ - [ 9ⁿ + n9^n-1 + n(n-1)9^n-2 ]

Total Base 10 numbers with length n - [numbers with no zeros + numbers with 1 zero + numbers with 2 zeros] 

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Total number of possible base 10 numbers of length n is 10ⁿ

1) numbers with no zeros = number of ways to fill n places with 9 digits 9ⁿ

its same as in binary n length so total 2ⁿ possible combinations 

2) numbers with 1 zero = number of ways to fill a zero in n places * number of ways to fill 9 digits in n-1 places
                                 n (9^n-1)       

3) number with 2 zeros = number of ways to fill 1 zero * number of ways to fill another zero * number of ways to fill 9 digits in n-2 places

                                n(n-1)9^n-2

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065 = 65

length of number is 2 isnt?

i consider valid numbers. otherwise all is same.

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@Anu

it is not told different numbers

right?

So, 65 and 065, 0065

all u have to considered differently

right?

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Yes ...

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i'm not understanding the doubt let n = 3

so there can be only one possible way to form a 65 which is with one zero 065

other wise if there are 2 zeros or no zeros or 3 zeros we cannot form 65

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I don't think first place 0 is allowed.

Otherwise, 065,0065,000065... all these numbers has same value but interpreted differently. There is no meaning then?

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 Ashwin Kulkarni n is fixed there can be one unique number with n fixed

if you are changing n then n=4 (0065), n=3(065), n=2 (65)

but the generalized formula 10ⁿ - [ 9ⁿ + n9^n-1 + n(n-1)9^n-2 ] for a specific n will give you a unique combination for 65

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maybe i should rephrase the question with How many base 10 numbers of length n are there with at least 3 zeros?

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why you guys are changing n? it is same as for binary number

let's consider n = 3, 

two = 010 and it is unique way out of 2³ combinations of 0 and 1

n = 4

two = 0010 and it is also unique out of 2⁴ possible combinations

Correct me if i am wrong

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I think you have phrased and understood the question correctly and the correct way to do it that Mk Utkarsh mentioned.

10ⁿ - [ 9ⁿ + n9^n-1 + n(n-1)9^n-2 ]

But the last term should have n(n-1)/2 and not n(n-1).

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why?

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We can chose any 2 of the n places for 0 in nC2 ways which is equal to n(n-1)/2. So shouldn't it be n(n-1)/2?

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absolutely correct :)

10ⁿ - [ 9ⁿ + ⁿC₁9^n-1 + ⁿC₂ 9^n-2 ]

thanks

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@Ashwin

yes that is correct

what is ur ans then?

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@ Mk Utkarsh
if there is any option?
really that would be helpful for ans
Actually arrarngement of 2 0's also not matter for calculation

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Why total number of n digit (base 10) numbers = $10^n$??
Shouldn't it be $9*10^{n-1}$   ???

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srestha  watch this from 16:20 to 22:00 shai explained the answer

https://www.youtube.com/watch?v=44xj31MZSKUhttps://www.youtube.com/watch?v=44xj31MZSKU

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yes, u mean 1st position select in 9 way and rest position select 10 ways

Correct one.

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Rishabh Gupta 2 

number of 1 digit 10 base numbers are 10 :P(pretty obvious)

10¹ = 10 and

9 * 10⁰ is 9

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@Utkarsh What about 2 digit numbers?

You can say that's 100. But this 100 includes 10 1-digit numbers(0 through 9) and 90 2-digit numbers(10 to 99).

So, when you say n-digit number. $10^n$ will include 1-digit numbers, 2- digit numbers ...., n-digit numbers.

In the question you fixed the digits to be n.

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So, when you say n-digit number. 10ⁿ will include 1-digit numbers, 2- digit numbers ...., n-digit numbers

yeah correct but 10ⁿ also includes 2 digit base 10 number(with 0 as first digit) and other numbers till 99

that adds upto 100.

01 is a 2 digit base 10 number and a (single digit) whole number