Question

If the letters of the BRING are permuted in all possible ways and the words thus formed are arranged as in dictionary order. Then find the ${59}^{th}$ word.

Answer 1

Hint: We will be using the concept of permutation and combination to solve the problem. We will first find the number of words starting with B and G, then we will further find the words starting with I and then will search the words starting with I to find the ${59}^{th}$ words.

Complete step-by-step answer:
Now, we have been given all the permutations of BRING. We have to find the ${59}^{th}$ word when we arrange all the permutations in dictionary order.
Now, we have to find the word starting with B. Now, we know the possible permutation of 4 things are 4!. Therefore, if we reserve the first word as B then the remaining 4 letters will be left and we have to find all possible permutations which are 4! = 24.
Now, similarly the number of words starting with G are 4!.
Now, we have total words in dictionary order till now as 48. So, in the next 24 words which start with I the ${11}^{th}$ word is the ${59}^{th}$ word.
Now, we have the number of words starting with IB as 3! = 6.
So, till now we have a total 48 + 6 = 54 words and we are left with 5 more. So, now we have the number of words starting with IGB as 2! = 2.
The words starting with IGN as 2! = 2.
So, till now we have 58 words, so the ${59}^{th}$ word will be IGRBN which starts with IGR and has the rest two letters in alphabetical order.
So, the ${59}^{th}$ word is IGRBN.

Note: To solve these types of questions it is important to approach the problem step wise like we have first found the letter of all words starting with B and so on. This simplifies the solution and shortens it also.