GATE CSE 2001 | Question: 2.1

Question

How many $4$-digit even numbers have all $4$ digits distinct?

• A. $2240$
• B. $2296$
• C. $2620$
• D. $4536$

Comments

Why all distinct four digit even numbers != all distinct four digit odd numbers ?

We are not applying any condition then why both of them have different count ?

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@ash_khola, 

That's a good question.

Here, we're applying some restrictions (distinct digits) on 4 digit numbers. That's makes, number of odd and even numbers with the given restrictions are not equal. 

 

How we can see the difference?

5 odd digits,  5 even digits.

Classify them as {1, remaining 4 odd digits }, {0, remaining 4 even digits}

 

In the set, for remaing 4 digit portion at the end of the number,

No.of 4 digit odd numbers where each digit is distinct = number of 4 digit even numbers where each digit is distinct 

 

Remaing is {1} and {0}.

No.of 4 digit odd numbers ending with 1 where each digit is distinct != number of 4 digit even numbers ending with 0 where each digit is distinct.

8x8x7 != 9x8x7

The difference is (9-8)(8×7) = 56.

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Lets make 2 cases

case_1
zero at last place

No. of possible choices 9*8*7*1=505
case_2

zero not ar last place but zero is not allowded at first place becz of 4 digit number
8*8*7*4=1792
 

totoal case = 1792+504 = 2296
Option B is correct

 

Answer 1

• If the number ends with a $0$ then there are $9$ choices for the first digit, $8$ for the second and $7$ for the third, which makes $1\times9\times8\times7= 504$ possibilities.

• If the number is even ending with something else than $0$ then there are $4$ choices for the last digit, $8$ choices for the first digit (no $0$ nor the last digit), $8$ for the second digit and $7$ for the third digit, which makes $4\times8\times8\times7 =1792$

Together, this gives $2296$ numbers with $4$ distinct digits that are even. Note that this does not allow leading $0$, as you see to want it based from the question.

Correct Answer: $B$

Comments

@abir_banerjee yeah afterwards I came to know as I continued with combinatorics lecture, thank u for replying.

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yeah got it

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how the option b is true ? how can we find 4 distinct number ? i didn't understand

 

Answer 2

$Method\ 1:$

$-$	$-$	$-$	$0$	$9.8.7.1$
$-$	$-$	$-$	$2$	$8.8.7.1$
$-$	$-$	$-$	$4$	$8.8.7.1$
$-$	$-$	$-$	$6$	$8.8.7.1$
$-$	$-$	$-$	$8$	$8.8.7.1$

$9.8.7.1+(8.8.7.1)\times 4=2296$

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$Method\ 2:$

$1)\ Total\ \#\ of\ 4\ digit\ numbers\ and\ all\ distinct=9.9.8.7=4536$

$2)\ \#\ of\ 4\ digit\ odd\ numbers\ and\ all\ distinct=8.8.7.5=2240$

$Even=Total-Odd=4536-2240=2296$

Comments

great one

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लय भारी

Answer 3

Available digits are { 0,1 , 2  ... 9}

Even { 0 2 4 6 8}

Odd { 1 3 5 7 9 }

Pos 4 must be one of even number . But Pos 1 can be odd or even but not 0.

(case 1 )

POS 1: ODD { ONE OF 1 3 5 7 9}

POS 2:7C1

POS:3 8C1                                                 SO Total is : 5 X (7x8x5)=1400

POS 4: ONE OF EVEN NO i.e. 5C1

(CASE 2: POS1 IS EVEN INLUDING 0)

POS 1: EVEN { ONE OF 0,2,4,6,8}

POS 2:7C1

POS:3 8C1                                                 SO Total is : 5 X (7x8x4)=1120

POS 4: ONE OF EVEN NO EXCLUDING THE DIGIT OF POS1 i.e. 4C1

(Case 3: POS1 IS 0)

POS 1: 0

POS 2:7C1

POS:3 8C1                                                 SO Total is : 1 X (7x8x4)=224

POS 4: ONE OF EVEN NO EXCLUDING 0  i.e. 4C1

The ans is : case1 + case2 - case3 =(1400+1120-224)=2296

Comments

Hi sir, I can't understood in this method .

  Why we take 3cases  sir your answer right but  I don't understand the case 1 .I agree with puja dii ans

Answer 4

I have tried in this way -

No of 4 digit even numbers= Total no of 4 digit even numbers -  Number having 0 at the front

   Total no of 4 digit even numbers =last place is filled in 2,4,6,8,0 ie 5 ways. Next place in 9 ways, next in 8 ways and next in 7 ways (go from right)

Number having 0 at the front = last place is filled in 2,4,6,8 ie 4 ways, next place in 8 ways and next in 7 ways and first place is for 0.

Required number of numbers = (7*8*9*5)-(7*8*4)  = 2520-224 = 2296

Comments

Last digit must be something among 0,2,4,6,8.

1 point u must note here is that it is 1st choosing and then araanging .

Now there is 2 possibility

1st 1-  if 0 is at last position , if so,

Then the other 3 positions can be filled in 9p3 ways that is equal to 9*8*7

Now here comes the 2nd case

Here take care that 1st digit is not zero otherwise it won't make 4 digits .

We have already considered the case where last digit is 0 so now in this case last digit can be filled in 4 ways possibilities may be 2,4,6,8 means total 4

2nd and 3rd digit can b filled in 8p2 ways .

And first digit in 8 ways

So it becomes 4*8*8*7

So total it becomes 4*8*8*7+9*8*7

Answer 5

Simple solution for this problem

Answer 6

Here in given question  we are find out 4 digit  distinct  number from the  decimal digit  0,1,2,3,4,5,6,7,8,9,

here see in above digit sequence  0 is included then we deal with  0 in deifferent manner if   they involved in  number  

first we find those number in which 0 is not involved

------                                  ------                                                   ------                          -------

    6                                         7                                                     8 {remaining}   4{2,4,6,8 possible}

 now deal  with zero number

---------                  ----------                -----------           ----------------

      7                          8                                9                     0{ here zero placed }

------------             -------------            --------------             ----------------

     7                           8                     0 {placed }                     4

------------                    --------                  ---------            -----------

           7                        0                                 8                        4  

---------                    ------------             ------------            ---------------

0 not possible      

if msb is  0 then number is not four digit

 

so total number =6*7*8*4+7*8*9+7*8*4+7*8*4=2296