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

@Chhotu comment explanation,

Same question could be done in one more way.

(All four digit no.) - (All four digit no. that are not even).

 

= 9*9*8*7 – 8*8*7*5

= 2296 

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@Kiyoshi , your method is exactly not correct ,

it can be written in a more appropriate way , 

(all distinct four digit numbers) – (All distinct four digit numbers that are not even) 

= 9*9*8*7 – 8*8*7*5

= 2296

 

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simple and easy  

find  all four digit number which digit is distinct and it is should be even

unit digit should be even  5cases

so overall possibilty is 5*9*8*7=2520

subtract those digit which msb is zero it  is should be  4*8*7=224

sooverall=2296

Answer 1

Simple solution for this problem

Answer 2

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

Answer 3

Here 0 creates trouble for us

So first let us ignore it

WE have 9 numbers {1,2,3,4,5,6,7,8,9}

_ _ _ _ fill these 4 digits in order to get an even number : 9*8*7*4 we all know this one

 

Now we will only count the cases where 0 is present

 

_ _ _ 0  example 1420 this is a valid number, 8250 this is also valid

so we are clearly able to see there are a lot of such numbers which are 9*8*7 --------(1)

 

now fix 0 in any of the remaining 2 places all of those will give same answer  X _ _ X ( as u cant put 0 in MSB)

like imagine a number 8024 this is a valid number (remember it has to be even also)

so 2 cases, hereby multiplying with 2 as the number is of the form

NUM | 0 | NUM | EVEN

8*7*4 *2

 

sum all the cases

we will get 2296

Answer 4

2296 is the correct answer.