Question

How many 4 letter words can be formed using the letters in the word MADHURI if
A) Letters can be repeated
B) Letters can’t be repeated

Answer 1

The word MADHURI contains seven letters and we are to choose 4 letter words out of 7 letter words. This is done by $\mathop C\nolimits_4^7$ no. of possible ways. Now we have chosen 4 letters and now if we have to arrange these letters that have been chosen. So if repetition is allowed then it can be done in ${4^4}$ way and if not allowed then in $4!$ ways. This will help to solve this question.

Complete step-by-step answer:
Here given a word ”MADHURI” and you are to pick any 4 letter and make the words as much as possible and then provide the number of total possible words.So as we know, if we choose $n$ number from $m$ numbers then its no. of ways are $\mathop C\nolimits_n^m$ that means choosing $n$ numbers from $m$ numbers.

(A) Now if Repeated is allowed which means like you all are given the word MADHURI and we have to choose four letters and repetition is also allowed which means MMMM , MMAD etc. are also the possible cases.
So here “MADHURI” contains 7 letters and all are distinct. So choosing 4 letter from “MADHURI”
Then, number of ways $= \,$ $\mathop C\nolimits_4^7$
Now arranging that 4 letter within itself where repetition is allowed $= {4^4}$
So total number of ways to choose 4 letters and arranging it $= \,$ $\mathop C\nolimits_4^7$ $\times$ ${4^4}$
$= \dfrac{{7!}}{{4!\left( 3 \right)!}} \times {4^4}$
$= \,\dfrac{{7 \times 6 \times 5 \times 4!}}{{4! \times 3 \times 2 \times 1}} \times {4^4}$
$= \,35 \times {4^4}$
$= \,8960$

(B) If it is saying that repetition is not allowed for example - we can’t take MMMM, we can take only one word one time.
So here also have to choose 4 letter from word MADHURI
Then, number of ways $= \,$ $\mathop C\nolimits_4^7$
Now arranging that 4 letters where repetition is not allowed $= 4!$
So the total number of ways to choose 4 letters and arranging it $= \;\mathop C\nolimits_4^7 \times 4!$
$= \dfrac{{7!}}{{4!\left( 3 \right)!}} \times 4!$
$= \dfrac{{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{3 \times 2 \times 1}}$
$= \,840$

Note: In general If repetition is allowed then arrangement of n letters within itself is given by $n^n$ ways.If repetition is not allowed then arrangement of n letters is given as $n!$ ways.