How many positive integers less than 2*10^4 are there in which each di

20000 (5 digits)
No number can be formed of 5 digits as 1 is not a prime number and at First place only Prime number 2 can come

single digit prime numbers are
2,3,5,7
4 digit numbers with prime numbers will be
So 4^4 = 256
3 digit numbers with prime numbers will be
4^3 = 64
2 digit numbers with prime numbers will be
4^2 = 16
1 digit numbers with prime numbers will be
4
Total = 256 + 64 + 16 + 4 = 340

C

Numbers less than 20,000 will have 5 digits or less.
This is a generic representation of this number: _ _ _ _ _
Clearly, we can't have a prime number for the 1st place since that would take the value more than 20k. Therefore, we start with 2nd place and so on. We can have only 2,3,5,7 as single digit prime numbers filling in the places.
So for a 4 digits number: we have 4 options for each digit. This gives us \(4 . 4 . 4 . 4 = 256\) possible numbers.
And for a 3 digits number: we have 4 options for each digit. This gives us \(4 . 4 . 4 = 64\) possible numbers.
And for a 2 digits number: we have 4 options for each digit. This gives us \(4 . 4 = 16\) possible numbers.
And for a 1 digit number: we have 4 options for the digit. This gives us \(4 . 1 = 4\) possible numbers.
Adding all possible numbers, we get 340. Hence C.

Integers less than 20000 in which each digit is a prime number : 2,3,5,7,22,23,25,27,32,33,35,37......

possible options
Single digit -> 4
2 digit -> 4 *4 = 16
3 digit -> 4*4*4 = 64
4 digit -> 4*4*4*4 =256
5 digit :
10000 till 20000 -> But here as 1st digit is always 1 which is not prime.. we cannot consider numbers between 10000 and 20000

Total : 4+16+64+256 = 340

Answer: C

is this the correct method
2* 10^4=20,000
single digit 2,3,5,7
two digit = 4*4 ie 16
three digit numbers 4*4*4= 64
four digit =4*4*4*4=256
we have reached 10,000
as tenth thousnadths digit is 1 all numbers beyond will have this number(1x,xxx) as non prime which makes them null and void
also the range 20,000 forbids us to consider 22,222
so we have to consider
256+64+16+4=340
Am i correct

_ - 4 ways
_ _ - 4 x 4 = 16 ways
_ _ _ - 4x4x4 = 64 ways
_ _ _ _ - 4x4x4x4= 256ways total = 340

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In other words, "How many positive integers less than 20,000 are there in which each digit is a prime number?"
The prime digits are: 2, 3, 5 and 7
Notice that, using the digits 2, 3, 5 and 7, we cannot create a 5-digit number that's less than 20,000
So, we must consider 4 possible cases: 4-digit numbers, 3-digit numbers, 2-digit numbers, and 1-digit numbers

4-digit numbers
There are 4 options for the first digit (2, 3, 5 or 7), 4 options for the second digit, 4 options for the third digit, and 4 options for the last digit
TOTAL number of 4-digit numbers = (4)(4)(4)(4) = 256

3-digit numbers
There are 4 options for the first digit (2, 3, 5 or 7), 4 options for the second digit, and 4 options for the last digit
TOTAL number of 3-digit numbers = (4)(4)(4) = 64

2-digit numbers
There are 4 options for the first digit (2, 3, 5 or 7), and 4 options for the last digit
TOTAL number of 2-digit numbers = (4)(4) = 16

1-digit numbers
There are 4 options: 2, 3, 5, 7

ANSWER = 256 + 64 + 16 + 4 = 340 = C

Cheers,
Brent

Numbers less than 20000, in which each digit is a prime number

There will be 4 prime numbers 2 3 5 7

_ _ _ _ => 4*4*4*4 = 216
_ _ _ => 4*4*4 = 64
_ _ => 4*4= 16
_=> 4

216 + 64 + 20
340

C

1 digit 4
2 digit 4*4 = 16
3 digit 4*4*4 = 64
4 digit 4*4*4*4 =256
Total : 4+16+64+256 = 340

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