Question

The letters of the word ‘MASTER’ are permuted in all possible ways and the words thus formed are arranged as in a dictionary. The rank of the word STREAM is:
(a) 597
(b) 480
(c) 612
(d) 385

Answer 1

Hint: Find out the number of words starting with A, E, M, R, SA, SE, SM, SR, STA, STE, STM and STRA. Then the next word will be ‘STREAM’. So after finding out the number of words before ‘STREAM’ using permutation and combinations, add 1 to the result to get the rank of ‘STREAM’.

Complete step-by-step answer:
Rank of a word means that we are supposed to find the position of that word when all permutations of the word are written in alphabetical order. Follow the steps to find the rank of any word:
Step I: Write down the letters in alphabetical order.
Step II: Find the number of words that start with a superior letter.
Step III: At last add 1 to the total number of words obtained to get the rank of the required word.
Now, let us come to the question. We have to find the position of ‘STREAM’.
Number of words that start with A $ =5! $ , as the remaining 5 letters can be arranged in $ 5! $ ways.
Number of words that start with E $ =5! $ , as the remaining 5 letters can be arranged in $ 5! $ ways.
Number of words that start with M $ =5! $ , as the remaining 5 letters can be arranged in $ 5! $ ways.
Number of words that start with R $ =5! $ , as the remaining 5 letters can be arranged in $ 5! $ ways.
Number of words that start with SA $ =4! $ , as the remaining 4 letters can be arranged in $ 4! $ ways.
Number of words that start with SE $ =4! $ , as the remaining 4 letters can be arranged in $ 4! $ ways.
Number of words that start with SM $ =4! $ , as the remaining 4 letters can be arranged in $ 4! $ ways.
Number of words that start with SR $ =4! $ , as the remaining 4 letters can be arranged in $ 4! $ ways.
Number of words that start with STA $ =3! $ , as the remaining 3 letters can be arranged in \[3!\] ways.
Number of words that start with STE $ =3! $ , as the remaining 3 letters can be arranged in \[3!\] ways.
Number of words that start with STM $ =3! $ , as the remaining 3 letters can be arranged in \[3!\] ways.
Number of words that start with STRA $ =2! $ , as the remaining 2 letters can be arranged in \[2!\] ways.
Now, the next word will be ‘STREAM’. Therefore, the rank of this word
 $ \begin{align}
  & =5!+5!+5!+5!+4!+4!+4!+4!+3!+3!+3!+2!+1 \\
 & =4\times 5!+4\times 4!+3\times 3!+2!+1 \\
 & =4\times \left( 5\times 4\times 3\times 2\times 1 \right)+4\times \left( 4\times 3\times 2\times 1 \right)+3\times \left( 3\times 2\times 1 \right)+\left( 2\times 1 \right)+1 \\
 & =4\times 120+4\times 24+3\times 6+2+1 \\
 & =480+96+18+3 \\
 & =597 \\
\end{align} $
Hence, the correct answer is option (a).

Note: Since no letter is appearing more than once, so we do not have to divide the number of words by any factorial number. Also note that, 1 is added in the last because all the steps above are to find the number of words that will appear before ‘STREAM’, so ‘STREAM’ will appear after word 596, that is, at place 597.