Question

A word has 4 identical letters and others different. If the total numbers of words that can be made with the letters of the word be 210 then the number of different letters in the word are
(A) 3
(B) 5
(C) 4
(D) 7

Answer 1

We solve this problem by first assuming the total number of letters in the word as $n$. Then we consider the formula for the number of ways of arranging $n$ objects with $r$ identical objects, $\dfrac{n!}{r!}$. Then we use this formula to find the number of words that can be formed from the letters given and then equate it to 210. Then we solve the obtained equation to find the value of $n$. Then we find the number of different letters in the word by subtracting the number of identical objects from it.

Complete step-by-step solution
We are given that a word has 4 identical letters and others are different.
We are also given that the total number of words that can be made with the letters of this word are 210.
Let us assume that the number of letters in the word is $n$. Then the number of different letters in the word is $n-4$ because 4 letters are identical.
So, we have that arranging the letters of the word with $n$ different letters and 4 identical letters we can make 210 words.
Now let us consider the formula, number of ways of arranging $n$ objects with $r$ identical objects is equal to
$\dfrac{n!}{r!}$
Using this formula, we get that number of ways of arranging the $n$ letters with 4 identical letters is equal to
$\Rightarrow \dfrac{n!}{4!}$
As we are given that this number is equal to 210, let us equate them. Then we get,
$\begin{aligned} & \Rightarrow \dfrac{n!}{4!}=210 \\\ & \Rightarrow n!=4!\times 210 \\\ & \Rightarrow n!=4!\times 30\times 7 \\\ & \Rightarrow n!=4!\times 5\times 6\times 7 \\\ & \Rightarrow n!=7! \\\ \end{aligned}$
So, we get that
$\Rightarrow n=7$
So, we get that the number of letters in the word are 7.
As we need to find the number of different letters in the word, that is equal to
$\begin{aligned} & \Rightarrow n-4 \\\ & \Rightarrow 7-4 \\\ & \Rightarrow 3 \\\ \end{aligned}$
Hence the answer is Option A.

Note: There is a possibility of one making a mistake while solving this problem, after finding the value of $n$, one might not subtract the 4 identical letters from it and answer the question as Option D. But here $n$ is the value of the total number of letters not the number of different letters.