Question

The letters of the word SURITI are written in all possible orders and these words are written out as in a dictionary. Let k be the rank of the word SURITI. Find sum of digits of k.

Answer 1

Hint: We will solve the question by using the concept of factorial which is a sub part of permutations. We are using it so that the orders of the letters also get included. We will first form an alphabetical order because the word is in the dictionary and after that we will find how many words are going to be formed out of each word. For the rank k we will add these numbers and solve the question.

Complete step-by-step answer:
According to the question we need to consider the word SURITI and consider its rank as k. We have to do that in a dictionary. For the word to be in the dictionary we will write it in alphabetical order. So the word now changes to IIRSTU.
Now we will consider the words that start with I. By this we will be able to find the ways in which the words starting with I only are formed. For this we will fill in the gaps of _ _ _ _ _ _. Now the first letter is I. Thus, we have I _ _ _ _ _ where I is showing 1 choice only. After this the second place is filled by any of the 5 letters and since repetition is not allowed therefore, we get I 5 _ _ _ _. Similarly by filling up the rest of the letters we will get I 5 4 3 2 1. By using multiplication in between them we will get $1\times 5\times 4\times 3\times 2\times 1$ which further results into 120.
Similarly we will find the letters starting with R. This can be done as R _ _ _ _ _.After filling the rest of the choices result into R 5 4 3 2 1. Thus we get $1\times 5\times 4\times 3\times 2\times 1$ = 120. But in this case we can see the repetition of the letter I. Since, it is repeated two times therefore we will divide 120 by 2 !. Thus we will get $\dfrac{120}{2}=60$. So, here we have 60 ways of getting the words starting from R.
Now, words beginning with SI are S I _ _ _ _ or S I 4 3 2 1. Therefore we get $1\times 1\times 4\times 3\times 2\times 1=24$ words. Also, words beginning from SR is given by $\dfrac{4!}{2!}=\dfrac{4\times 3\times 2\times 1}{2\times 1}$ where, dividing by 2! will delete the repetition of I. Thus we get here 12 words. Moreover, the words starting with S T are $\dfrac{4!}{2!}=\dfrac{4\times 3\times 2\times 1}{2\times 1}$ which results in 12 words.
Now, we will consider the words beginning with S U I as S U I _ _ _. And the number is given by 3! Or 6. After this we are left with two words in alphabetical order which are SURIIT and SURITI. Since, these two words form 1 way each in the dictionary. So, by adding all the numbers we will get 120 + 60 + 24 + 12 + 12 + 6 + 1 + 1 = 236.
Hence, the value of k is 236.

Note: We are given that the word is in the dictionary and along with its rank that is why we have taken the word as alphabetical order here. Use of factorial is also one of the reasons to be in the solution so that we find the words by considering their all respected orders. As the word also carries two I’s as repetition so to get rid of the repetition we will divide the permutations by 2! in which the I’s are being repeated. As we are supposed to find the rank, we have decided to first find out the number of all the words that can be formed before the required word SURITI and after that we will find it’s rank by adding all the numbers. By following these points one will get to the right answer.