Question

How many words can be formed by taking $4$ letters at a time from the letters of the word ‘MORADABAD’?

Answer 1

Hint: In the word MORADABAD there are two letters that come more than one time. In this word, ‘A’ comes thrice and the letter ‘D’ comes twice. So here we will take all the cases in which $4$ letters word can be formed from the letters of the word ‘MORADABAD’.

Complete Step-by-step Solution
Given: The word ‘MORADABAD’ is given and we have to find all the possible cases in which $4$ letters word can be formed by this word.
First, we will take the word ‘MORADABAD’ in which $6$ different letters are used that is: M-$1$ time
O-$1$ time, R-$1$ time, A-$3$ times, B-$1$ time, D-$2$ times.
We will find out all the possible cases of word-formation.
Case 1: Here we will discuss the word-formation cases in which all the letters are different.
So all four letters comes different in ${}^6{C_4} \times 4!\,$ ways
i.e. $
   = \dfrac{{6 \times 5}}{{2!}} \times 4 \times 3 \times 2 \\
    \\
$
$ = 360$ ways
Case 2: Here we will discuss in which $2$ alike of one kind and $2$ alike of another kind of letters can come.
So two alike letters of one kind and two alike letters of other kind can come in ${}^2{C_2} \times \dfrac{{4!}}{{2!2!}}$ ways
i.e. $ = \dfrac{{4 \times 3 \times 2 \times 1}}{{2 \times 1 \times 2 \times 1}}\, = 6$ ways.
Case 3: Here we will discuss in which two letters are alike and two letters are different.
So two alike and two different letters can come in ${}^2{C_1} \times {}^5{C_2} \times \dfrac{{4!}}{{2!}}$
$ = \dfrac{{2 \times 5 \times 4}}{2} \times \dfrac{{4 \times 5 \times 2 \times 1}}{{2 \times 1}} = 240$ ways.
Case 4:Here we will discuss in which three letters alike and one different come.
So three alike of one kind & one letter different can come in ${}^1{C_1} \times {}^5{C_1} \times \dfrac{{4!}}{{3!}}$ ways
i.e. $5 \times 4 = 20$ ways.
Now we will add all the ways of all the cases.
Hence total possibilities $ = 360 + 6 + 240 + 20$
$ = 626$ ways.

626 words can be formed by taking $4$ letters at a time from the letters of the word ‘MORADABAD.

Note:
In this problem first, we will take the word ‘MORADABAD’ in which $6$ different letters are used that is: $m - 1$ time O-$1$ time, R-$1$ time, A-$3$ times, B-$1$ time, D-$2$ times. Like the letter, ‘A’ comes thrice and the letter ‘D’ comes twice and we required $4$ letters out of six different letters so we found all the cases.