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    find out the distinct three-letter words that can be formed using the word singapore.

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    Find the number of three letter words that can be formed by using the letters of the word 'MASTER'

    Click here👆to get an answer to your question ✍️ Find the number of three letter words that can be formed by using the letters of the word 'MASTER'

    Find the number of three letter words that can be formed by using the letters of the word

    Question ′ MASTER ′

    This question has multiple correct options

    A

    120

    B

    6 P 3 ​

    C

    6 C 3 ​

    D

    720

    Easy Open in App

    Updated on : 2022-09-05

    Solution Verified by Toppr

    Correct options are A) and B)

    There are six distinct digits "MASTER"

    Number of three letter =

    6 C 3 ​ ×3!= 6 P 3 ​ =120

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    Find out the distinct three letter words that can be formed using the word SINGAPORE

    AnswerVerified Hint: In this question, we need to consider all the possibilities. Now, calculate the number of 3 letter words that can be formed ...

    Find out the distinct three letter words that can be formed using the word SINGAPORE

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    Hint: In this question, we need to consider all the possibilities. Now, calculate the number of 3 letter words that can be formed with all the letters being different using the permutation formula given by \[{}^{n}{{P}_{r}}\]. Then find the number of words having two letters similar using the combinations given by the formula \[{}^{n}{{C}_{r}}\] and then arrange them. Then find the words with all the 3 letters the same and add all these to get the result.

    Complete step-by-step answer:

    Now, from the given word SAHARANPUR in the question we have

    S, A, A, A, H, R, R, N, P, U

    Now, we need to find the 3 letter words that can be formed using these letters

    Let us first consider the case that all the 3 letters to be different

    Now, we have to arrange the 7 different letters in 3 places

    As we already know that arrangement of n things in r places can be done using the permutations given by the formula

    \[{}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}\]

    Now, on comparing with the formula we have

    \[n=7,r=3\]

    Now, on substituting the respective values we get,

    \[\Rightarrow {}^{7}{{P}_{3}}\]

    Now, this can be further written as

    \[\Rightarrow \dfrac{7!}{\left( 7-3 \right)!}\]

    Now, on further simplification we get,

    \[\begin{align}

    & \Rightarrow 7\times 6\times 5 \\

    & \Rightarrow 210 \\

    \end{align}\]

    Thus, 210 3- letter words can be formed with all letters being different

    Now, let us find the number of 3-letter words having two letters same

    Here, the possible two same letters can be either A or R

    As we already know that selection can be done using the combinations given by the formula

    \[{}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}\]

    Now, we can select either of A or R in

    \[\Rightarrow {}^{2}{{C}_{1}}=\dfrac{2!}{1!1!}\]

    That means in 2 ways

    Now, the next letter can be any of the remaining 6 letters which can be done in

    \[\Rightarrow 6\text{ ways}\]

    Now, the arrangement of these letters with 2 letters can be done in

    \[\Rightarrow \dfrac{3!}{2!}\]

    \[\Rightarrow 3\text{ ways}\]

    Now, the 3-letter words that can be formed with two letters same is given by

    \[\Rightarrow 2\times 6\times 3\]

    Now, on further simplification we get,

    \[\Rightarrow 36\]

    Thus, there are 36 3- letter words with 2 letters same

    Now, we need to find the number of letters with three letters same

    Here, A is the only letter that is repeated thrice

    So, the word can be AAA which has only 1 way

    Now, the total number of 3-letter words that can be formed from the given word are

    \[\Rightarrow 210+36+1\]

    Now, on simplifying it further we get,

    \[\Rightarrow 247\]

    Hence, the correct option is (c).

    Note:

    It is important to note that we need to consider the words having 2 letters the same and three letters the same with the words having all letters different because these all satisfy the given condition. Here, neglecting any of the cases gives the incorrect option.

    It is also to be noted that in the case that we considered having 2 letters to be the same after choosing the letters we need to arrange them because we are finding the words that are possible.

    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.

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    Find out the distinct four-letter words that can be formed using the word SINGAPORE. * 256 1024 3024 2048

    Find out the distinct four-letter words that can be formed using the word SINGAPORE. * 256 1024 3024 2048

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