Question

How many different ways can the letter of the word PROPORTION be arranged?

Answer 1

758 ways

word proportion can be arranged in 758 ways .

We have two P’s, two R’s, three O’s, and all the others, i.e., T, I, and N have appeared once.

Now, the following cases arise:

Words with four distinct letters.

We have 6 letters in total, i.e,(I,N,P,R,O and T) so we can arrange this letters in (64)×4!=360 ways.

2. Words with exactly a letter repeating twice.

We have P, R, and O repeating itself. Now one of these three letters can be chosen in (31)=3 ways.

The other two distinct letters can be selected in (52)=10 ways.

Now each combination can be arranged in 4!2!=12 ways.

So total no. of such words =3×10×12=360 .

3. Words with exactly two distinct letters repeating twice.

Two letters out of the three repeating letters P, R, and O can be selected in (32)=3 ways.

Now each combination can be arranged in 4!2×2!=6 .

So total no. of such words =3×6=18 .

4. Words with exactly a letter repeating thrice.

We have only one option for this as our main letter that is O.

Now we have to select 1 letter out of the 5 remaining options so no. of ways to this =(51)=5 .

Now each combination can be arranged in 4!3!=4 .

So total no. of such words =1×5×4=20 .

Thus all possible no. of arrangements =360+360+18+20=758 ways.

Step-by-step explanation:

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