Question

How many four-letter words can be formed by using the letters of the word “HARD WORK”?

Answer 1

Hint: Here, we will find the number of four-letter words that can be formed where the letter R comes at most once, that is each letter comes once. Then, we will find the number of four-letter words that can be formed where the letter R comes twice. Finally, we will add the two results to get the number of four-letter words that can be formed by using the letters of the word “HARD WORK”.
Formula Used:
The number of permutations in which a set of $$n$$ objects can be arranged in $$r$$ places is given by $${}^n{P}_r={\displaystyle \frac{n!}{\left(n-r\right)!}}$$, where no object is repeated.
The number of permutations to arrange $$n$$ objects is given by $${\displaystyle \frac{n!}{r_1!r_2!\dots r_n!}}$$, where an object appears $$r_1$$ times, another object repeats $$r_2$$, and so on.

Complete step-by-step answer:
The number of letters in the word ‘HARD WORK are 8, where R comes twice.
The letters are to be arranged in 4 places.
The number of permutations in which a set of $$n$$ objects can be arranged in $$r$$ places is given by $${}^n{P}_r={\displaystyle \frac{n!}{\left(n-r\right)!}}$$, where no object is repeated.
The number of permutations to arrange $$n$$ objects is given by $${\displaystyle \frac{n!}{r_1!r_2!\dots r_n!}}$$, where an object appears $$r_1$$ times, another object repeats $$r_2$$, and so on.
Thus, we can find the answer using two cases.


Case 1: The letter R is not repeated in the 4 places.
We have 7 letters to be placed in 4 spaces.
The 7 letters are H, A, R, D, W, O, K.
We observe that no letter is being repeated.
Substituting $$n=7$$ and $$r=4$$ in the formula $${}^n{P}_r={\displaystyle \frac{n!}{\left(n-r\right)!}}$$, we get
$${}^7{P}_4={\displaystyle \frac{7!}{\left(7-3\right)!}}={\displaystyle \frac{7!}{4!}}={\displaystyle \frac{7\times 6\times 5\times 4\times 3\times 2\times 1}{4\times 3\times 2\times 1}}=840$$

Therefore, the number of four-letter words that can be formed where the letter R comes at most once, is 840.
Case 2: The letter R is repeated in the 4 places.
In the 4 places, 2 places will be taken by the two R’s, and the remaining 2 places will be taken by any of the remaining 6 letters.
The number of ways in which this is possible can be found by using combinations.
Therefore, we get
Number of four-letter words where R is repeated (order not important) $$={}^2{C}_2\times {}^6{C}_2$$
Since the order matters in the number of words we need to find, we will find the order in which the 4 letters (chosen in $${}^2{C}_2\times {}^6{C}_2$$ ways) can be placed in the 4 places, where R is repeated.
This can be found by using the formula $${\displaystyle \frac{n!}{r_1!r_2!\dots r_n!}}$$.
Thus, the four chosen letters can be ordered in $${\displaystyle \frac{4!}{2!}}$$ ways.
Therefore, we get the number of four-letter words where the 7 letters are placed in 4 places, and R is repeated, is given by $${}^2{C}_2\times {}^6{C}_2\times {\displaystyle \frac{4!}{2!}}$$ ways.
Here, $${}^2{C}_2\times {}^6{C}_2$$ is the number of ways of choosing the letters to be placed within the 4 places, and $${\displaystyle \frac{4!}{2!}}$$ is the number of ways in which the chosen 4 letters can be ordered.
Simplifying the expression, we get
Number of four-letter words where the letter R is repeated $$=1\times {\displaystyle \frac{6\times 5}{2\times 1}}\times {\displaystyle \frac{4\times 3\times 2\times 1}{2\times 1}}=180$$
Therefore, the number of four-letter words where the letter R is repeated is 180.
Finally, we will calculate the number of four-letter words that can be formed using the letters of the word “HARD WORK”.
The number of four-letter words that can be formed using the letters of the word “HARD WORK” is the sum of the number of four-letter words that can be formed where the letter R comes at most once, and the number of four-letter words that can be formed where the letter R comes twice.
Thus, we get the number of four-letter words that can be formed using the letters of the word “HARD WORK” is $$840+180=1020$$ words.
Therefore, 1020 four-letter words can be formed using the letters of the word “HARD WORK”.

Note: We used combinations to get the number of four-letter words where the letter R is repeated (order not important). The number of combinations in which a set of $$n$$ objects can be arranged in $$r$$ places is given by $${}^n{C}_r={\displaystyle \frac{n!}{r!\left(n-r\right)!}}$$. Therefore, the number of ways in which the 2 R’s can be placed in 2 places is $${}^2{C}_2$$, and the number of ways to place the remaining 6 letters in the 2 places is $${}^6{C}_2$$. By multiplying these, we get the number of ways to place the 8 letters in the 4 places, such that the letter R comes twice, and order of letters does not matter.