How many five-digit numbers?

[microke]

How many five-digit numbers are there, if the two leftmost digits are even, the other digits are odd and the digit 4 cannot appear more than once in the number?

A 1875
B 2000
C 2375
D 2500
E 3875

Dear Friends, Can any one explain this problem?

[kanak]

My approach:

Visualize the below empty slots

- - - - -

The first two slots need to be even numbers. The first one can be filled by 4 numbers (2, 4, 6, 8). The second can be filled by 4 numbers (0, 2, 6, 8) i.e., exclude 4 and include 0 in the second spot.

Hence, the first two can be filled up in 4 x 4 = 16 ways.

The remaining three slots have to be filled up using any of 1, 3, 5, 7, 9 digits as they can be repeated. i.e.,

third slot filled by 5 digits
fourth by 5
and fifth by 5

So, 5 x 5 x 5 = 125 ways of filling the last three slots.

Combining, we have 125 x 16 = 2000.

B is the answer.

[microke]

Thanks for the explanation. I misunderstood the left most digit as right-most digits and included zero in the place..But I came across this question in www.youdowetest.com and the answer was C.

[smarty80in]

The best answer is C.

Not considering the fact that 4 cannot appear more than once, we have a total of 4*5*5*5*5=2500. Now we deduct the possibilities where 4 does appear more than once (in this case it can appear only twice on the two leftmost even digits). In order to do so, we put 4 in the first and second leftmost digits. The rest of the digits are odd: 5*5*5=125.
2500-125=2375

[debarshi7]

smarty , are you considering 0 as an even integer while filling up the
second position from the left

[smarty80in]

yes 0 is an even integer
You can search "Is 0 an even integer" on google and you will get the answer

[microke]

Thanks smarty80in! You are right...this approach is simple and effective......

and zero is a perfect even number!

[Carol]

smarty80in, I didn't understand why you subtracted 125 from 2500.

[younggun044]

Correct answer is "C".
Smarty approach is absolutely fine. Goodone!!

[Sunny22uk]

As per O.G. , 0 is neither a postive integer nor a negative integer.

[microke]

Then we should follow only GMAC advice....

[smarty80in]

yes, but still 0 is an even integer

See the page number 108 of OG11
it has given definition and list of even integer
and the OG11 list includes clearly includes 0

Any integer that is divisible by 2 is an even integer, the set of even integers is
{...,-4,-2,0,2,4,6,8,...}


My sincere advice to all you guys, please revise the basics properly.

[beeparoo]

As per O.G. , 0 is neither a postive integer nor a negative integer.

Your comment here is irrelevant. We are not concerned about negative or positive numbers. We are only talking about EVEN or ODD numbers.

[smarty80in]

subtracted 5*5*5 cases ( in which 4 is coming on 1st and 2nd place) from total number of cases.
We need to subtract because 4 is not allowed more than once and as 4 is an even integer and even integer is allowed only at first 2 places, so 4 can repeat only at first 2 places.