Question

The number of 4 letter words that can be formed using the letters of the word RAMANA.

Answer 1

Hint: Count the number of words in the given letter. As the combination of words matters in the given question, using the concept of combinations to find the number of 4 letter words that can be formed by the given word. We will use the formula, $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ to solve the combinations expressions.

Complete step by step Answer :

Let us consider the given words ‘RAMANA’. In the given word, we can see that there are, 1R, 3A’s, 1M, and 1N.

Now, we observe that for forming 4 letter words using these letters there are a few cases that we should consider.

Let us consider them one-by-one.

Case 1:- Let the first case be the word formed in which all the 4 letters are different.
In this case, the first blank will have 4 choices out of 1R, 1A, 1M, and 1N. As in this case, all the letters of the word are different, thus the second blank would have only 3 choices. Due to a similar reason, the third blank would have 2 choices and the first one will only be left with 1 choice.
$\Rightarrow {\text{Total possible words}} = 4! \\\ = 4 \times 3 \times 2 \times 1 \\\ = 24 \\\$

Case 2:- Let this case be the one in which the word contains two same letters, say 2A’s and 2 distinct letters.
In this case, two same letters are occurring, thus the total ways in which this can happen is, $\dfrac{{4!}}{{2!}}$.
Two distinct letters out of the remaining 3 letters (other than A) can be chosen in ${}^3{C_2}$.
$\Rightarrow {\text{Total possible words}} = {}^3{C_2} \times \dfrac{{4!}}{{2!}} \\\ = 3 \times \dfrac{{24}}{2} \\\ = 36 \\\$

Case 3:- Let this case be the one in which the word contains three same letters, say A’s and 1 distinct letter.
In this case, three same letters are occurring, thus the total ways in which this can happen is, $\dfrac{{4!}}{{3!}}$.
One distinct letter can be chosen in ${}^3{C_1}$ ways.
$\Rightarrow {\text{Total possible words}} = {}^3{C_1} \times \dfrac{{4!}}{{3!}} \\\ = 3 \times \dfrac{{24}}{6} \\\ = 12 \\\$

Now, we will add the value obtained in all three cases to find the total of 4 letter words formed by the word RAMANA.
$\Rightarrow 24 + 36 + 12 = 72$

Thus, the number of 4 letter words that can be formed using the letters of the word RAMANA are 72.

Note: In the questions where you need to find the words formed by the letter given to you, count the number of letters in the word, then the number of distinct and same letters. Based on the nature of the required word, you need to carefully distribute or divide the possibilities into various cases. Make sure to consider all of them and not miss any. Also, you need to be clear about the combination concept and factorial concept.