Question

Number of ways of selecting four letters from proportion

Answer 1

First of all see that which letters are repeating.

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

Now,the cases are:-

1.Words with four distinct letters.

We have 6 letters all total,i.e,(I,N,P,R,O and T)so we can arrange this letters in (6C4)×4!=360

2.Words with exactly a letter repeated twice.

We have P,R and O repeating itself.Now one of this three letter can be choose in (3C1)=3ways.

The other two distinct letters can be selected in (5C2)=10ways

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

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

3.Words with exactly two distinct letters repeated twice.

Two letters out of the three repeating letters P,R and O can be selected in (3C2)=3ways

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 repeated 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 =(5C1)=5

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

So,total no. of such words=1×5×4=20

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