Question

What is the number of four letter words that can be formed by taking the letters of the word KANYAKUMARI?
(a) 2218
(b) 1680
(c) 1708
(d) 1714

Answer 1

Hint: To solve this problem we will split it into four parts i.e. first when all four letters are different, second when three letters are alike, third when 2 are alike and rest two are alike with each other, fourth when 2 are alike and 2 are different. We will find the number of four letter words that can be formed and then add them all to find the required answer.

Complete step by step answer:
We are given the word KANYAKUMARI,
And we have to find the number of four letter words that can be formed using the above word’s characters.
We know that in the word KANYAKUMARI there are 2 K's and 3 A's and 6 different letters N, Y, U, M, R, I.
So to solve this we will divide this problem into four cases,
Case 1: When all the four letters are different,
In this case word can be formed using the letters N, Y, U, M, R, I, K, A
So we get
Number of words will all 4 letters different as = $^{8}{{C}_{4}}=1680$
Case 2: When there are 3 alike letters in four letter word,
There is only one case like that i.e. when all three ‘A’ are used in the word. And one letter is chosen from rest 7 letters
So we get number of words with 3 letters alike and 1 different as = $^{7}{{C}_{1}}\times \dfrac{4!}{3!}=28$
Case 3: When 2 letters alike of one type and two are alike of another type
In this case we can form word using the letters 2 ‘K’ and 2 ‘A’
 So we get number of words with 2 letters alike of one type and 2 alike of another type as = $\dfrac{4!}{2!\times 2!}=6$
Case 4: When only 2 letters are alike
In this case we can assume the alike letters to be either 2 ‘K’ or 2 ‘A’
So we get number of words in this case as = $^{2}{{C}_{1}}{{\times }^{7}}{{C}_{2}}\times \dfrac{4!}{2!}=504$
Hence we get total number of 4 letter words that can be formed using the letters of the word KANYAKUMARI as,
$=1680+28+6+504=2218$

So, the correct answer is “Option A”.

Note: You need to think carefully before start solving this question as you may skip all the cases and assume the $^{8}{{C}_{4}}$ as the answer which is also present as option (b) but that will be wrong. So you have to make cases whenever you are solving questions related to permutation that involve similar letter words.