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 \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\], where no object is repeated.
The number of permutations to arrange \[n\] objects is given by \[\dfrac{{n!}}{{{r_1}!{r_2}! \ldots {r_n}!}}\], 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 \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\], where no object is repeated.
The number of permutations to arrange \[n\] objects is given by \[\dfrac{{n!}}{{{r_1}!{r_2}! \ldots {r_n}!}}\], 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 \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\], we get
\[{}^7{P_4} = \dfrac{{7!}}{{\left( {7 - 3} \right)!}} = \dfrac{{7!}}{{4!}} = \dfrac{{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{4 \times 3 \times 2 \times 1}} = 840\]

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 \[\dfrac{{n!}}{{{r_1}!{r_2}! \ldots {r_n}!}}\].
Thus, the four chosen letters can be ordered in \[\dfrac{{4!}}{{2!}}\] 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 \[{}^2{C_2} \times {}^6{C_2} \times \dfrac{{4!}}{{2!}}\] 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 \[\dfrac{{4!}}{{2!}}\] 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 \[ = 1 \times \dfrac{{6 \times 5}}{{2 \times 1}} \times \dfrac{{4 \times 3 \times 2 \times 1}}{{2 \times 1}} = 180\]
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 \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]. 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.