# The letters of the word 'SURITI' are written in all possible orders and these words are written out as in a dictionary.

Question:

The letters of the word 'SURITI' are written in all possible orders and these words are written out as in a dictionary. Find the rank of the word 'SURITI'.

Solution:

In a dictionary, the words are arranged in the alphabetical order. Thus, in the given problem, we must consider the words beginning with I, I, R, S, T and U.

I will occur at the first place as often as the ways of arranging the remaining 5 letters, when taken all at a time.

Thus, I will occur 5! times.

Similarly, R will occur at the first place the same number of times.

∴ Number of words starting with  I = 5!

Number of words starting with $R=\frac{5 !}{2 !}$

The word will now start with S, which is as per the requirement of the word SURITI.

Alphabetically, the next letter would be I, i.e. SI. The remaining four letters can be arranged in 4! ways.

Alphabetically, the next letter would now be R, i.e. SR. The remaining four letters can be arranged in $\frac{4 !}{2 !}$ ways.

Alphabetically, the next letter would now be T, i.e. ST. The remaining four letters can be arranged in $\frac{4 !}{2 !}$ ways.

Alphabetically, the next letter would now be U, i.e. SU, which is as per the requirement of the word SURITI.

After SU, alphabetically, the third letter would be I, i.e. SUI. Thus, the remaining 3 letters can be arranged in 3! ways.

The next third letter that can come is R, i.e. SUR, which is as per the requirement of the word SURITI.

After SUR, the next letter that will come is I, i.e. SURI, which is as per the requirement of the word SURITI.

The next word arranged in the dictionary will be SURIIT.

Then, the next word will be SURITI.

Rank of the word SURITI in the dictionary $=5 !+\frac{5 !}{2 !}+4 !+\frac{4 !}{2 !}+\frac{4 !}{2 !}+3 !+2=236$