Question

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

Answer 1

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.