# In how many ways can the letters of the word MULTIPLE be arranged in the following cases -?

(i) without changing order of the vowels,

(ii) keeping position of the vowels fixed

(iii) without changing the relative order of vowels and consonants.

### 2 Answers

- PuzzlingLv 76 months ago
There are 8 letters, but L appears 2 times. So without restriction, there are 8!/2! = 20,160 ways to arrange the letters.

The problem with this question is I'm not 100% sure of their intended meaning for each of the restrictions. I'll add my assumptions and you can check them against what you think they mean.

PART 1:

If I'm understanding the question, they want any arrangement, but U, I and E must be in that order somewhere in the arrangement.

In any of the possible arrangements (without restriction), the vowels could appear equally in one of 3! = 6 orders. So if we divide by 6, we'll have the number of ways that U, I, E will still be in that order.

8! / (2! 3!)

= 40,320 / 12

= 3,360 ways

PART 2:

Now they want the vowels to be in their current positions (2, 5 and 8) but they could be rearranged (3! = 6 ways). Then the consonants could placed in the remaining 5 positions in 5! / 2! = 60 ways (don't forget the repeated Ls)

3! * 5!

= 6 * 60

= 360 ways

PART 3:

Here the vowels must be U,I,E and the consonants M,L,T,P,L

Just imagine 8 positions and pick the places for the 3 vowels. There is 1 order for them in those chosen positions and 1 order for the consonants that go in the other positions. So we just need to calculate "8 choose 3". Equivalently we could pick the 5 positions for the consonants then place the vowels. The value of "8 choose 5" gives the same result.

C(8,3) = C(8,5)

= (8 * 7 * 6) / (3 * 2 * 1)

= 56 ways