4 digit roll number formed by using digits 0 to 9 without repetition,

Question

4 digit roll number formed by using digits 0 to 9 without repetition, are to be assigned to students of a school. A roll number should neither start with 0 or end with an odd digit. If the number of roll numbers which can be formed this way is 500 more than the number of students in the school, then what is the number of students in the school?

A. 1296

B. 1500

C. 1796

D. 1976

E. 1476

A. 1296

B. 1500

C. 1796

D. 1976

E. 1476

chetan2u · Accepted Answer

Responding to a PM..
How does 9*8*7*5 come..
Restrictions
a) even number
b) does not start with 0..

Let's take first restriction..
a) even number..
So first fill up the units digit, it can be any of 0,2,4,6,8 --- 5 ways...
Now fill up remaining 
first digit- any of remaining 9 except the DIGIT at units place
Second and third - remaining 8 and 7..
So total ways - 9*8*7*5

BUT this contains the one starting with 0..
So let's take restriction 2..
b) does not start with 0..
How many start with 0..
So first place in only one way that is 0..
Again fill up units digit- 4 ways ( all even digits except 0)
Second and third- remaining 8 and 7
Total ways - 2*8*7*4

So our answer 9*8*7*5-1*8*7*4...

chetan2u · Answer

Without restriction of 0 at first place...
9*8*7*5
Numbers with 0 at first place..
1*8*7*4

Answer = 9*8*7*5-8*7*4=(8*7)(45-4)=56*41=2296...

So number of students=2296-500=1796
C

vyascd · Answer

we have 4 digits _ _ _ _
now we can not put 0 in the first place so 9 9 8 7 ------------- This represents all 4 digit number
now odd numbers will be exactly half of this number so we are left with 9*9*8*7 / 2 --------------- This will give us 9*9*4*7 which is somewhere near 2200
Subtracting 500 from the number should give us a number near 1700 hence the answer

If there is something wrong with this solution let me know

Thanks

sheshadre · Answer

There's a slight mistake. 9*9*8*7 doesn't represent all 4 digit numbers but only those 4 digit numbers with unique digits. For eg. 9*9*8*7 does not include 7777 or 6666. Hope this helps

Fuf · Answer

Without restriction of 0 at first place it should be 
9*8*7*5? why not 10*9*8*5?

mbeneduzi · Answer

How I've approached:

All possible outcomes - (possible outcomes in the first restriction + possible outcomes in the second restriction - the overlap of outcomes with both restrictions) - 500:

All possible outcomes- 10*9*8*7

Outcomes in the first restriction (zero as the first digit) - 9*8*7
Outcomes in the second restriction (odd number as the last digit) - 5*9*8*7

Sum of restricted outcomes 9*8*7*6 - but in both of them we have the condition  0, Number, Number, Odd Number , so we have to take out the overlap (make this condition be real just once). 1(0 as the first digit)*5 (odd as the last digit)*8*7

10*9*8*7 - (9*8*7*6 - (8*7*5)) - 500
8*7(10*9 - (9*6 - 5)) - 500
8*7 (90 - 54 + 5) - 500
8*7(41) - 500 = 1796

Hope it helps!

pulkit1080 · Answer

My Approach :

Restrictions: 
1. First digit should not be 0 (9 possibilities: 1,2,3,4,5,6,7,8,9)
2. Last digit should not be odd (5 possibilities 0,2,4,6,8)

Lets take 2 Cases, when 0 is the last digit and when 0 is not the last digit
Case 1: When 0 is the last digit
4th digit - 0 (1 possibility)
1st digit - Any number 1-9 (9 possibilities)
2nd digit - Any remaining number out of 10 after 2 numbers are selected (8 possibilities)
3rd digit - Any remaining number out of 10 after 3 numbers are selected (7 possibilities)
Hence possible cases become  9  *  8  *  7  *  1  = 504

NOTE : First fill the possibilities where restrictions are applied

Case 2: When 0 is not the last digit
4th digit - 2,4,6,8 (4 possibilities)
1st digit - Any number from 1-9 except 2,4,6,8 (8 possibilities since only one of 2,4,6,8 will be used at a time)
2nd digit - Any remaining number out of 10 after 2 numbers are selected (8 possibilities)
3rd digit - Any remaining number out of 10 after 3 numbers are selected (7 possibilities)
Hence possible cases become  8  *  8  *  7  *  4  = 1792

Total becomes = 1792 + 504

Since number of students are 500 less, hence number of students = 1792 + 504 - 500 = 1796

Hope it helps!

EshaFatim · Answer

chetan2u

"Numbers with 0 at first place..
1*8*7*4"

could you please explain why isn't it 1*9*8*4? With zero in the first place, there remains 9 other digits to work with in the next one, no?
if anyone else could help a swell, pleas do.

TanyaBhargava · Answer

why can't we take it in single go, without cases? im confused.
i have done 8*7*8*5
-> 5 because we can take 5 even, then first 8, because we can't take 0 and 1 even digit, has gone to units place.
then 8 and 7 like we normally do. i dont understand the error here

Sandesh013 · Answer

Bunuel 
how is it not 2020, because there are 10 numbers to start with from 0-9, then if we apply the first condition i.e should not start with 0 we will have 9*8*7 and then the last digit cannot be odd so it will be 9*8*7*5 which results into 2520 subtracting 500 it is 2020 which is not in the options.
Please post your response.

Bunuel · Answer

For the calculation 9 * 8 * 7 * 5, the issue is that if the first three digits include even numbers, the options for the last digit won't be 5.

Similarly, in 8 * 7 * 6 * 5, if 0 is chosen as the last digit, there would be 9 options for the first digit, not 8.

In these types of questions, you should start by evaluating the digits with restrictions first to avoid mistakes. Also, since the value of the last digit determines the available options for the other digits, we should split the cases and calculate them separately:

The number of cases with 0 as the units digit + the number of cases with other even digits as the units digit =

= (1 option, 0, for the units digit ) * 9 * 8 * 7 + (4 options for the units digit ) * 8 * 8 * 7 = 2,296.

Hope it's clear.

OJA · Answer

Bunuel - I have a doubt here. For the part where we are counting the number of cases with other even digits as the units digit - shouldn't it be 8*8*7*4 ? Can you share the reasoning behind saying it should be 8*7*6*4 ?

Bunuel · Answer

You are right it should be (4 options for the units digit ) * 8 * 8 * 7.

rahumangal · Answer

we have 4 digits _ _ _ _
now we can not put 0 in the first place so 9 9 8 7 ------------- This represents all 4 digit number
now odd numbers will be exactly half of this number so we are left with 9*9*8*7 / 2 --------------- This will give us 9*9*4*7 which is somewhere near 2200
Subtracting 500 from the number should give us a number near 1700 hence the answer

If there is something wrong with this solution let me know

4 digit roll number formed by using digits 0 to 9 without repetition,

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4 digit roll number formed by using digits 0 to 9 without repetition,

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