Problem Solving

papgust wrote:In how many ways can the letters of the word "DOUBLE" be re-arranged such that the order in which the vowels appear in the word does not change?

Solution given is 6!/3!. Why are we dividing by 3! here? Please help me understand

uttam.albela wrote: Out of 6 places, select 3 places. No of ways = 6C3
No. of ways to Arrange these letters DBL in selected 3 places = 3!
Now we are left with 3 places where we need to put OUE in this same order.
No of ways to do this = 1

So total number of ways = 6C3 * 3! * 1 = 6!/3! = 120

Let me know if any part of it is not clear to you.

papgust wrote:

uttam.albela wrote: Out of 6 places, select 3 places. No of ways = 6C3
No. of ways to Arrange these letters DBL in selected 3 places = 3!
Now we are left with 3 places where we need to put OUE in this same order.
No of ways to do this = 1

So total number of ways = 6C3 * 3! * 1 = 6!/3! = 120

Let me know if any part of it is not clear to you.

Thanks for the post! I'm still not clear why we are doing 6C3 here. Otherwise i'm clear abt rest of the steps.

See we have 6 places and we need to select 3 place to arrange 3 non-vowels. So number of ways to choose 3 places out of 6 is 6C3. Once we have 3 seats for 3 non-vowels, we can arrange then in 3! ways.

My method:
Consider vowels first and this can be arranged in only 1 way. -- I agree vowels can be arranged in only one way, but only after we have selected the 3 places for vowels. 3 seats for vowels can be selected from 6 seats available in 6C3 ways. Is it clear to you now?
Then remaining three places can be arranged with 3 remaining letters in 3! ways. - This is correct.
Altogether is 3!*1 ways. Tell my why my understanding is wrong

In how many ways can you re-arrange the letters in the word "DOUBLE" such that first letter must start with "D" and last letter must end with "E"?

uttam.albela wrote: We have 6 letters in DOUBLE, so there are 6 places to be filled by these 6 letters..

Let us first think of arranging DBL.

Out of 6 places, select 3 places. No of ways = 6C3
No. of ways to Arrange these letters DBL in selected 3 places = 3!
Now we are left with 3 places where we need to put OUE in this same order.
No of ways to do this = 1

In how many ways can you re-arrange the letters in the word "DOUBLE" such that first letter must start with "D" and last letter must end with "E"?

papgust wrote:So you are saying that both approaches lead to same answer i.e. 6C3 * 3!.

But consider this qn,

In how many ways can you re-arrange the letters in the word "DOUBLE" such that first letter must start with "D" and last letter must end with "E"?

This answer is 4!. My qn is why are we not multiplying 6C4 with 4! here. Even in this qn, we can say that first we need to keep D and E and then re-arrange other 4 letters or first arrange 4 letters and then take D and E. In either case, according to you, we must multiply by 6C4 with 4!. But OA is only 4!.

Hope i'm making sense!