# Find the number o

There are 3N,4E,2D,1l,1P,1C

Since letters are repeating so we use the formula .

Total number of arrangements are:

= 1663200

(i) If word starts with P.

Now P will be fixed,

We now need to arrange 11 letters.

Where there are 4E,3N,2D

Since letters are repeating so we use the formula .

No. of arrangements

= 138600

(ii) do all the vowels always occur together

We will consider vowels as a same letter. Here there are 5vowels i.e IEEEE,

So, these can be arranged in ways as E is repeating 4 times.

Now total letters are 7+1 = 8

Now there are 3N,2D in remaining letters.

These can be arranged in ways.

Total ways

= 16800

iii) do all the vowels never occur together

Vowels never occur together = Total arrangements – vowels occur together

= 1663200 – 16800

= 1646400

iv) do the words begin with I and end in P?

Let’s fix I and P at the ends.

There are 2D,4E,3N.

As letters are repeating,

Total arrangements

= 12600

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