A 5-digit identification code is to be created as a sequence that cont

Question

A 5-digit identification code is to be created as a sequence that contains each integer in the set {1, 2, 3, 4, 5} exactly once. If all identification codes are possible except those containing even integers next to each other, how many different identification codes are possible?

A. 120
B. 96
C. 72
D. 48
E. 24

Let's just do Total minus evens next to each other and call it a day:

chetan2u · Accepted Answer

Total ways to create 5-digit identification code = 5!

Restriction: Both even number cannot be together.

When both the even digits are taken together as one digit, we have 4 digits, that is 1, 3, 5 and (24) => 4! to arrange 4 digits and 2 & 4 can be arranged in 2! ways within themselves. Hence, 4!*2

Ways we are looking at = 5!-4*2! = 120-48 = 72

C

Bunuel · Answer

That calculation gives the number of ways when the two even numbers, 2 and 4, are next to each other. By treating them as one unit, we have a total of 4 units: {2, 4}, {1}, {3}, {5}. These 4 units can be arranged in 4! ways, and 2 and 4 within their unit can be arranged in 2! ways, giving the total number of ways when 2 and 4 are next to each other as:
 
4! * 2! = (4 * 3 * 2 * 1) * 2 = 48.
 
Since the total number of ways to arrange 5 different digits is 5! = 120, the number of codes possible except those containing even integers next to each other would be:
 
Total - Restriction = 120 - 48 = 72.
 
Answer: B.

Hope it's clear.

Archit3110 · Answer

let x be number 24 
x135
can be arranged in 24 ways
and 42 now be y
y135 can be arranged in 24 ways
total ways to arrange 5 digits is 120
120- ( 24+24) ; 72
option C

ruis · Answer

Is there another approach to this question? How would we solve it using the Permutation formula?

MartyMurray · Answer

A 5-digit identification code is to be created as a sequence that contains each integer in the set {1, 2, 3, 4, 5} exactly once. If all identification codes are possible except those containing even integers next to each other, how many different identification codes are possible?

A. 120
B. 96
C. 72
D. 48
E. 24

Let's just do Total minus evens next to each other and call it a day:

We must create codes such that even integers are separated by odd integers.

First arrange the odd integers. There are 3 odd integers. So, they can be arranged in 3! = 6 ways.

Now, having arranged the odd integers, we have the following possible slots for the 2 even integers:

_ O _ O _ O _

We see that there are 4 slots we can choose for the even integers.

So, for example, we can choose the following:

2 O 4 O _O _

Thus, we are choosing 2 out of 4 slots. So, the ways to chose the slots are 4C2 = 6.

There are 2 ways to order the 2 even integers in the 2 slots they occupy. So, we have 6 × 2 = 12 ways to arrange the even integers.

So, with 6 ways to arrange the odd integers and 12 ways to arrange the even integers, we have 6 × 12 = 72 total posssible arrangements.

A. 120
B. 96
C. 72
D. 48
E. 24

Correct answer:  C

DanTheGMATMan · Answer

Let's just do Total minus evens next to each other and call it a day:

https://www.youtube.com/watch?v=EHFqsA3KXm8

daynesj24 · Answer

Can you explain how you got the 4 * 3 * 2 * 2

nosaj · Answer

Hi  chetan2u ,

This makes sense and thank you for the explanation. I just wanted to test my understanding.

If for example the question was {1,2,3,4,5,6} for a 6-digit code - would the answer be 6! * (4! * 3!) - because there are 6 ways to arrange the 3 even digits?

Thanks in advance!

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

A 5-digit identification code is to be created as a sequence that cont

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

GMAT Problem Solving (PS) Questions

A 5-digit identification code is to be created as a sequence that cont

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia