A five digit number is to be formed using each of the digits

Question

A five digit number is to be formed using each of the digits 1, 2, 3, 4 and 5 ONLY ONCE. How many numbers can be formed when 1 and 2 are not together ?

(A) 48
(B) 36
(C) 72
(D) 60
(E) 120

MacFauz · Accepted Answer

No. of ways the 5 digit number can be formed = 5! = 120

Let us fuse 1 & 2.
No. of ways in which 1 and 2 are together = 4! x 2! (Multiplying by 2! because 1 and 2 can be arranged in 2! ways) = 24*2 = 48

No. of ways 1 & 2 wont be together = 120 - 48
=72

Bunuel · Answer

Great solution. +1.

Just to elaborate the case when 1 and 2 are together:

Glue 1 and 2 together and consider them as one unit: {12}.

Now, 4 units {12}, {3}, {4}, and {5} can be arranged in 4! ways and 1 and 2 within their unit can be arranged in 2 ways ({12} or {21}), thus total = 4!*2.

arakban99 · Answer

Awesome solutions MacFauz and Bunuel, +1 kudos for both. Thanks a lot

manuelpalomares · Answer

Great learning on this one!! thank you very much!

arakban99 · Answer

No KUDOS? :evil:

Bunuel · Answer

Good questions deserves kudos as much as a good solution. +1.

MacFauz · Answer

Haha... I second that....

ramannanda9 · Answer

C

Just subtract from the total number of cases.

total number of combinations=5*4*3*2*1=120 
When two numbers are together, we will have pack the two numbers together, therefore combinations=4!*2(numbers can be arranged themselves)

The cases where they are not together=total-cases when they are together

120-48=72

adiagr · Answer

Reqd ways = (Total No. of ways - cases when both 1 and 2 are together)

5 numbers can be arranged in 5 places in 5! ways. Now tie 1 and 2 together so effectively there are 4 nos. they can be arranged in 4! ways. 1 and 2 can be arranged within themselves in 2!ways.

Reqd. Answer = 5! - 4!2! = 120 - (24*2) = 72

Answer: C

amanvermagmat · Answer

Another way to ensure that 1 and 2 are not together is to use 'Gaps' method. Basically, when we are given 'n' objects to be arranged so that 'a' out of them should NOT be together - we first arrange the remaining 'n-a' objects, and then arrange these 'a' objects in the gaps created by those 'n-a' objects.

I will try to explain by this question.

We have to arrange 1, 2, 3, 4, 5 so that 1 and 2 are not together. So lets first arrange the remaining numbers. 
Remaining 3 numbers can be arranged in 3! = 6 ways.

Now, for each of these 6 ways, the arranged numbers create 4 gaps:
...3...4...5...

one gap before 3, another gap between 3 and 4, another gap between 4 and 5, and the last gap after 5.

So for every arrangement of those 3 numbers, we have 4 gaps - and we can now place the numbers '1' and '2' in any two out of these 4 gaps (and thus they will not touch each other). Number of ways of doing that = 4P2 (arranging 4 objects at 2 places) = 12 ways.

Thus total ways = 3! * 4P2 = 6*12 = 72. Hence  C answer

PS - this same can be extended to any such question. So if we have 7 people out of which A,B,C have to be separated (no two of them can sit together), we can first arrange remaining 4 people in 4! ways, and now out of the 5 gaps created (4 people in a line will create 5 gaps), we can place A,B,C in any of the 3 gaps out of 5 in 5P3 ways (60). Thus total ways = 4! * 5P3 = 24 * 60 = 1440.

Nunuboy1994 · Answer

5!- 4!2!

C

rashwiniyer · Answer

Ans - C.
Total number possible - total number possible with 1 & 2 together = 
5! - (4! * 2!) = 120 - 48 = 72

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 five digit number is to be formed using each of the digits

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

My Rewards

GMAT Problem Solving (PS) Questions

A five digit number is to be formed using each of the digits

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

My Rewards