Combinatorics 01

[knight247]

How many even 4 digit numbers can be formed by using the digits 0-9 so that no two digits are repeated?
A 2296
B 2396
C 2444
D 2456
E 2486

Don't have an OA. Detailed explanations would be appreciated

[shankar.ashwin]

Its

9*9*8*7 = 4536 (for thousands, hundreds, tens and units place respectively)

You sure of options?

[krishnasty]

IMO A

This question can be solved in 2 parts.
part 1) when the unit digit is 0 (rememeber, any number ending with 0 is an even number)
9*8*7*1 = 504 ( 1 option for unit digit*9 digits for thousand digit, 8 for hundered and 7 for tens)

part 2) when the unit digit is any number from 2,4,6,8
8*8*7*4 = 1792( 4 option for unit digit, 8 for thousand,8 for hundered and 7 for tens)
tota; : 1792 +504 = 2296

[shankar.ashwin]

_ _ _ _

You know total Nos of possible 4 digit integers is 4536 (9*9*8*7)

To not consider separate cases including 0's, as the prev soln. Consider Nos of odd and subtract it with total, makes calc much easier.

For odd the units can be filled by (1,3,5,7 or 9)

_ _ _ 5

First can take 8 (exclude 0 and the last odd)
Second can take 8 (exclude first and last)
Third can take 7

So, 8*8*7*5 = 2240

Hence even = 4536-2240 = 2296

[Brent@GMATPrepNow]

How many even 4 digit numbers can be formed by using the digits 0-9 so that no two digits are repeated?
A 2296
B 2396
C 2444
D 2456
E 2486

For most counting questions we can break the task into stages and apply the Fundamental Counting Principle (FCP).

Stage 1: Select units digit
Stage 2: Select tens digit
Stage 3: Select hundreds digit
Stage 4: Select thousands digit

Two important things:
1) When you use the FCP, always begins with the most restrictive stage
2) To handle the problematic situation where having a zero in the thousands position is no good, let's first determine the total number of 4-digit integers that may or may not have a zero in this position, and then we'll subtract the number of 4-digit integers that have a zero in this position.

4-digit integers that may or may not have a zero in the thousands position
Stage 1: can be accomplished in 5 ways (even digits only)
Stage 2: can be accomplished in 9 ways (once we select a digit for stage 1, there are 9 digits to choose from)
Stage 3: can be accomplished in 8 ways
Stage 4: can be accomplished in 7 ways
Total number of 4-digit integers = 5x9x8x7 = 2520

4-digit integers that have a zero in the thousands position
We'll begin the most restrictive stage, which is stage 4.
Stage 4: can be accomplished in 1 ways (must be zero)
Stage 1: can be accomplished in 4 ways (even digit other than 0)
Stage 2: can be accomplished in 8 ways
Stage 3: can be accomplished in 7 ways
Total number of 4-digit integers with zero in thousands position = 1x4x8x7 = 224

Final answer = 2520 - 225 = 2296, so the answer is A

Cheers,
Brent